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Noncommutative Symmetric Functions and Permutation

Enumeration

A Dissertation

Presented to

The Faculty of the Graduate School of Arts and Sciences

Brandeis University

Department of Mathematics

Ira M. Gessel, Advisor

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

by

Yan Zhuang

May, 2018

The signed version of this signature page is on file at the Graduate School of Arts and

Sciences at Brandeis University.

This dissertation, directed and approved by Yan Zhuang’s committee, has been accepted and

approved by the Faculty of Brandeis University in partial fulfillment of the requirements for

the degree of:

DOCTOR OF PHILOSOPHY

Eric Chasalow, Dean of Arts and Sciences

Dissertation Committee:

Ira M. Gessel, Dept. of Mathematics, Chair.

Olivier Bernardi, Dept. of Mathematics

Bruce E. Sagan, Dept. of Mathematics, Michigan State University

c© Copyright by

Yan Zhuang

2018

Dedication

献给我的爸爸妈妈。

iv

Acknowledgments

My time as a Ph.D. student at Brandeis University has been a period of tremendous

personal growth and I owe thanks to the many individuals who have helped me along this

journey. To begin, I give my heartfelt gratitude to my advisor, Ira Gessel. I am blessed to

have an advisor who is not only a brilliant mathematician but also a patient, generous, and

supportive mentor. It has been a privilege working with you, Ira, and I cannot thank you

enough for your kind guidance. I am also grateful to the other two members of my dissertation

committee, Olivier Bernardi and Bruce Sagan. Olivier, I’ve learned so much combinatorics

from you at Brandeis, and Bruce, I owe much to you for your extensive feedback, advice, and

encouragement from afar.

My interactions with my professors, friends, and colleagues in the Brandeis mathematics

department have helped me become the person who I am today. I thank Susan Parker and

Becci Torrey for helping me become the best mathematics educator I can be, and for their

incredible support throughout my time at Brandeis. Susan and Becci, you were there for me

and believed in me during the darkest moments of my graduate student career, and I cannot

express how much that means to me. I thank Janet Ledda and Catherine Broderick for their

hard work providing administrative support for the department. I thank Ruth Charney for

supervising my minor exam and for being an inspirational mathematician. I thank Arunima

Ray for being a role model and a kind friend for us graduate students. I thank Jordan Awan,

Angelica Deibel, Joshua Eike, Eric Hanson, Devin Murray, and Jordan Tirrell for five years’

worth of fond memories and engaging conversations—we had some good times together! And

a very special shout-out to all of the amazing students who I’ve had the pleasure of teaching

(and learning from) at Brandeis—I’ve said this many times, but it’s worth repeating again: I

am so proud of all of you.

v

I’ve had many enlightening mathematical discussions with and received encouragement

from a number of people in the wider algebraic combinatorics community; in addition to

Bruce Sagan, individuals who I would like to thank in particular include Sami Assaf, Sara

Billey, Sergi Elizalde, Darij Grinberg, Brian Miceli, Kyle Petersen, Brendon Rhoades, and

Alexander Woo. I would be remiss to forget my professors from my undergraduate studies at

Goucher College—especially Justin Brody, Bernadette Tutinas, and Micah Webster—who

helped me to begin this journey by cultivating my passion for mathematics, and I am also

grateful to Anant Godbole for his REU program at East Tennessee State University that

introduced me to the joy of mathematical discovery.

Last but certainly not least, I thank my father 庄平 and my mother 曹晓瑛 for their

unconditional love and support. The many sacrifices that they have made for me are impossible

to enumerate—this work is dedicated to you.

vi

Abstract

Noncommutative Symmetric Functions and Permutation Enumeration

A dissertation presented to the Faculty of theGraduate School of Arts and Sciences of Brandeis

University, Waltham, Massachusetts

by Yan Zhuang

This Ph.D. dissertation is a compilation of material from four papers [23, 60, 61, 22] that

develop and apply methods involving noncommutative symmetric functions to permutation

enumeration, and in particular to the theory of descent statistics: permutation statistics that

depend only on the descent set and length of a permutation. We prove a generalization of

Gessel’s run theorem and use it to enumerate permutations with parity restrictions on peaks

and valleys, and to give a general method for enumerating permutations by descent statistics

that are expressible in terms of run lengths. Next, we prove new identities that express

Eulerian polynomials in terms of polynomials encoding the distribution of other descent

statistics (and vice versa)—including refinements of formulas previously found by Stembridge

and Petersen—and enumerate permutations by various descent statistics together with the

inversion number. Finally, we introduce the notion of a shuffle-compatible permutation

statistic and develop a theory of shuffle-compatibility for descent statistics, unifying previous

results of Stanley, Gessel, Stembridge, Aguiar–Bergeron–Nyman, and Petersen.

vii

Preface

One of the primary goals of permutation enumeration is to study the distributions of

permutation statistics. Many classical permutation statistics—including the descent set,

descent number, major index, peak set, and peak number—are based on the notion of descents.

More precisely, these are all what we call “descent statistics”: permutation statistics that

depend only on the descent set and length of a permutation. The study of descents and

descent statistics dates back to the work of Percy MacMahon [38], and many connections

have since been established between the study of descents and a variety of other subjects,

including theoretical computer science [34], discrete geometry [44], and genomics [15].

In their seminal 1995 paper [18], Israel Gelfand, Daniel Krob, Alain Lascoux, Bernard

Leclerc, Vladimir Retakh, and Jean-Yves Thibon introduced the Hopf algebra Sym of

noncommutative symmetric functions and elucidated its connections to combinatorics, repre-

sentation theory, Lie algebras, and mathematical physics. It is a noncommutative version

of the classical Hopf algebra of symmetric functions and is the graded dual of the Hopf

algebra QSym of quasisymmetric functions introduced by Ira Gessel [19]. However, it is worth

noting that noncommutative symmetric functions implicitly appeared earlier in the Ph.D.

dissertation of Gessel [24] in the context of permutation enumeration. Gessel showed that

many permutation enumeration formulas involving descents can be proven by first deriving a

lifting of the formula in Sym and then applying an appropriate homomorphism. Moreover,

he proved a result that we call the “run theorem”, which allows one to obtain noncommutative

symmetric function formulas counting permutations with restrictions on the lengths of their

increasing runs (i.e., distances between consecutive descents). Much of the work in this

present dissertation expands upon this theme set by Gessel, and so this dissertation can in

some ways be considered as a sequel to Gessel’s dissertation.

viii

The structure of this dissertation is as follows. Chapter 1 is an introduction to permutation

enumeration and the theory of descent statistics, and Chapter 2 is an introduction to some

relevant aspects of the theory of noncommutative symmetric functions. The material in these

two chapters consist of basic definitions and results as well as some technical lemmas and

propositions found in the present author’s papers referenced below.

Our main result in Chapter 3 is a generalization of Gessel’s run theorem which allows

for a much wider variety of restrictions on run lengths. We use the run theorem and our

generalization of the run theorem to find simple expressions for the exponential generating

functions for permutations with parity restrictions on peaks and valleys—thus answering

a question posed by Liviu Nicolaescu on discrete Morse functions arising in combinatorial

topology—and to give a general method for computing bivariate generating functions counting

permutations by descent statistics that are expressible in terms of run lengths. The proof of

Theorem 3.3 (a) is based on joint work with Ira Gessel [23], and the rest of the work in this

chapter is based on material from [60].

In Chapter 4, we prove a number of new identities expressing Eulerian polynomials in

terms of polynomials encoding the distribution of other descent statistics (and vice versa),

including refinements of formulas previously found by John Stembridge [58] and Kyle Petersen

[41, 42]. Here, we also find expressions for q-exponential generating functions that count

permutations by various descent statistics together with the inversion number. The work in

this chapter is based on material from [61].

Finally, in Chapter 5, we introduce and study the notion of a shuffle-compatible permu-

tation statistic. We define the shuffle algebra of a shuffle-compatible permutation statistic;

this algebra has a natural basis whose structure constants encode the distribution of the

statistic over shuffles of permutations. We prove a shuffle-compatibility criterion which

implies that the shuffle algebra of any shuffle-compatible descent statistic is a quotient of

ix

QSym, as well as a dual criterion which allows one to prove that a descent statistic is

shuffle-compatible by constructing a suitable subcoalgebra of Sym. These results are used to

prove that many descent statistics are shuffle-compatible and to give explicit descriptions of

their shuffle algebras, unifying past results of Richard Stanley [51], Gessel [19], Stembridge

[58], Aguiar–Bergeron–Nyman [2], and Petersen [42]. The work in this chapter is based on

material from [22], which is joint work with Ira Gessel.

The three appendices at the end of this dissertation contain six tables which summarize

information about the permutation statistics that appear in this body of work.

We note that not all the material from the four papers [23, 60, 61, 22] appear in this

dissertation; instead, we have chosen to highlight results obtained via noncommutative sym-

metric functions. Overall, we hope that this dissertation presents an accessible introduction

to permutation enumeration and the role of noncommutative symmetric functions in the

study of descent statistics.

x

Contents

Dedication iv

Acknowledgements v

Abstract vii

Preface viii

Table of Contents xi

Chapter 1. Introduction to permutation enumeration and descents 1

1.1. Permutations, descent sets, and compositions 1

1.2. Descent statistics 4

1.3. Possible values of some descent statistics 9

1.4. Alternating permutations, descents, and runs 12

Chapter 2. Noncommutative symmetric functions 15

2.1. Basic definitions 15

2.2. Homomorphisms on Sym 18

2.3. Several noncommutative symmetric function formulas 21

2.4. The bialgebra structure of Sym 30

2.5. Quasisymmetric functions 32

Chapter 3. The run theorem and its applications 37

3.1. Introduction 37

xi

3.2. Gessel’s run theorem 38

3.3. The generalized run theorem 40

3.4. Permutations with parity restrictions on peaks and valleys 44

3.4.1. Statement of main result 44

3.4.2. All peaks odd and all valleys even 45

3.4.3. All peaks and valleys even 47

3.4.4. All peaks and valleys odd 51

3.5. Counting permutations by run-expressible descent statistics 53

3.5.1. General setup 53

3.5.2. Counting by peaks (and variations) 55

3.5.3. Counting by double ascents (and variations) 58

3.5.4. Counting by biruns and up-down runs 62

Chapter 4. Eulerian polynomials and descent statistics 66


4.2. Descents of type B permutations 69

4.3. Several new Eulerian polynomial identities 75

4.4. Main results 77

4.4.1. On peaks and descents 77

4.4.2. On left peaks and descents 81

4.4.3. On up-down runs and descents 84

4.5. Two remarks: the inverse major index and alternating analogues 88

Chapter 5. Shuffle-compatible permutation statistics 90


5.2. Shuffle algebras 92

5.2.1. Definition and basic results 92

xii

5.2.2. Basic symmetries yield isomorphic shuffle algebras 95

5.2.3. A note on Hadamard products 99

5.3. Theory of shuffle-compatibility for descent statistics 100

5.3.1. Shuffle-compatibility of Des, Pk, and Lpk 100

5.3.2. A shuffle-compatibility criterion for descent statistics 102

5.3.3. A dual shuffle-compatibility criterion for descent statistics 104

5.3.4. Monoidlike noncommutative symmetric functions 107

5.3.5. Monoidlike elements and shuffle-compatibility 109

5.4. Explicit descriptions of shuffle algebras 112

5.4.1. Shuffle-compatibility of pk and (pk, des) 112

5.4.2. Shuffle-compatibility of lpk and (lpk, des) 116

5.4.3. Shuffle-compatibility of udr and (udr, des) 117

5.4.4. Shuffle-compatibility of des and (des,maj) 121

5.5. Non-shuffle-compatible permutation statistics 123

5.6. Open problems and conjectures 124

5.7. Two remarks: the Malvenuto–Reutenauer algebra and the descent algebra 125

Appendix A. Summary of permutation statistics 128

Appendix B. Summary of permutation statistic equivalences 132

Appendix C. Summary of shuffle-compatible permutation statistics 133

Bibliography 135

xiii

CHAPTER 1

Introduction to permutation enumeration and descents

1.1. Permutations, descent sets, and compositions

We begin by reviewing some basic material from permutation enumeration relating

to descents. Let π = π1π2 · · · πn be a permutation in Sn, the set of permutations of

[n] = {1, 2, . . . , n}, which are called n-permutations.1 Also, let |π| be the length of π, so that

|π| = n whenever π ∈ Sn. We say that i ∈ [n − 1] is a descent of an n-permutation π if

πi > πi+1. The set of descents, or descent set, of π is denoted Des(π), so that

Des(π) := { i ∈ [n− 1] : πi > πi+1 }.

Any i ∈ [n− 1] which is a not a descent of π is called an ascent of π.

Every permutation can be uniquely decomposed into a sequence of maximal increasing

consecutive subsequences—or equivalently, maximal consecutive subsequences containing no

descents—which we call increasing runs. For example, the descents of π = 85712643 are 1, 3,

6, and 7, and the increasing runs of π are 8, 57, 126, 4, and 3. It is clear that the lengths of

the increasing runs of a permutation determine its descent set, and vice versa.

Sometimes it is more convenient to represent a descent set of an n-permutation with a

composition of n which encodes the lengths of its increasing runs. Given a subset S ⊆ [n− 1]

with elements s1 < s2 < · · · < sj, let Comp(S) be the composition (s1, s2 − s1, . . . , sj −

sj−1, n − sj) of n, and given a composition L = (L1, L2, . . . , Lk), let Des(L) := {L1, L1 +

L2, . . . , L1 + · · ·+ Lk−1} be the corresponding subset of [n− 1]. Then, Comp and Des are

inverse bijections. If π is an n-permutation with descent set S ⊆ [n−1], then we call Comp(S)

1By convention, we take S0 to consist of only the empty word.

1

CHAPTER 1. INTRODUCTION TO PERMUTATION ENUMERATION ANDDESCENTS

the descent composition of π, which we also denote by Comp(π). Note that the descent

composition of π gives the lengths of the increasing runs of π. Conversely, if π has descent

composition L, then its descent set Des(π) is Des(L).

We partially order compositions of n by reverse refinement, that is, L = (L1, . . . , Lk)

covers M if and only if M can be obtained from L by replacing two consecutive parts Li and

Li+1 with Li + Li+1. For example, we have (7, 6) < (1, 2, 4, 5, 1). Note that if L and M are

descent compositions of n-permutations, then L ≤ M if and only if Des(L) ⊆ Des(M); in

other words, Comp and Des are order-preserving bijections.

Given a composition L, we let l(L) denote the number of parts of L, let |L| denote the

sum of the parts of L, and let L � n indicate that |L| = n (i.e., L is a composition of n). If

L = (L1, . . . , Lk) is a composition of n, we write(nL

)for the multinomial coefficient

(n

L1,...,Lk

)and we write

(nL

)qfor the q-multinomial coefficient(

n

L1, . . . , Lk

)q

:=[n]q!

[L1]q! [L2]q! · · · [Lk]q!

where

[n]q! := (1 + q)(1 + q + q2) · · · (1 + q + · · ·+ qn−1).

An inversion of an n-permutation is a pair of indices (i, j) with 1 ≤ i < j ≤ n such that

πi > πj. Then the number of inversions of π is denoted inv(π). For example, the inversions

of π = 1432 are (2, 3), (2, 4), and (3, 4), so inv(π) = 3. It is well known that the polynomial

counting n-permutations by inversion number is given by the nth q-factorial, i.e.,∑π∈Sn

qinv(π) = [n]q!. (1)

Lemma 1.1. Let L be a composition of n. Then:

(a) The number of n-permutations with descent composition K ≤ L—or equivalently,

with descent set contained in Des(L)—is the multinomial coefficient(nL

).

2


(b) The polynomial counting n-permutations with descent composition K ≤ L—or

equivalently, with descent set contained in Des(L)—by inversion number is the q-

multinomial coefficient(nL

)q. That is,∑π∈Sn

Comp(π)≤L

qinv(π) =

(n

L

)q

.

See [55, Examples 2.2.4 and 2.2.5] for proofs. This result on counting n-permutations

with a descent set contained in a prescribed set can then be used to count those with a

prescribed descent set.

Lemma 1.2. Let L be a composition of n. Then:

(a) The number β(L) of n-permutations with descent composition L—or equivalently,

with descent set Des(L)—is given by the formula

β(L) =∑K≤L

(−1)l(L)−l(K)

(n

K

). (2)

(b) The polynomial

βq(L) :=∑π∈Sn

Comp(π)=L

qinv(π)

counting n-permutations with descent composition L—or equivalently, with descent

set Des(L)—by inversion number is given by the formula

βq(L) =∑K≤L

(−1)l(L)−l(K)

(n

K

)q

.

The proof of Lemma 1.2 is immediate from Lemma 1.1 and the inclusion-exclusion

principle. Part (a) of Lemmas 1.1 and 1.2 were originally due to MacMahon [38], whereas

part (b) of these lemmas were due to Stanley [52].

Finally, we define three involutions on permutations given by symmetries: reversion,

complementation, and reverse-complementation. Given π = π1π2 · · · πn, we define the reversal

3


πr of π to be

πr := πnπn−1 · · · π1,

the complement πc of π to be the n-permutation obtained by (simultaneously) replacing

the ith smallest letter in π with the ith largest letter in π for all 1 ≤ i ≤ n, and the

reverse-complement πrc of π to be πrc := (πr)c = (πc)r. For example, given π = 136254, we

have πr = 452631, πc = 641523, and πrc = 325146.

1.2. Descent statistics

A permutation statistic is a function defined on the set⋃∞n=0 Sn of all permutations.

The descent set Des and the inversion number inv are important examples of permutation

statistics, and here is a list of some other permutation statistics that we will study in this

dissertation.

• The descent number des. The descent number des(π) of π ∈ Sn is defined to be

des(π) := |Des(π)|,

i.e., the number of descents of π.

• The major index maj. The major index maj(π) of π ∈ Sn is defined to be

maj(π) :=∑

k∈Des(π)

k,

i.e., the sum of the descents of π.

• The peak set Pk and peak number pk. We say that i (where 2 ≤ i ≤ n− 1) is a peak

of π ∈ Sn if πi−1 < πi > πi+1. The peak set Pk(π) of π is defined to be

Pk(π) := { 2 ≤ i ≤ n− 1 : πi−1 < πi > πi+1 }

and the peak number pk(π) of π to be

pk(π) := |Pk(π)|.

4


• The valley set Val and valley number val. We say that i (where 2 ≤ i ≤ n− 1) is

a valley of π ∈ Sn if πi−1 > πi < πi+1. Then Val(π) and val(π) are defined in the

analogous way.

• The left peak set Lpk and left peak number lpk. We say that i ∈ [n − 1] is a left

peak of π ∈ Sn if i is a peak of π or if i = 1 and is a descent of π. Thus, left peaks

of π are peaks of 0π shifted by 1. The left peak set Lpk(π) of π is the set of left

peaks of π and the left peak number lpk(π) of π is the number of left peaks of π.

• The right peak set Rpk and right peak number rpk. These are defined in the same

way as the left peak statistics, except that right peaks of π are peaks of π0.

• The exterior peak set Epk and exterior peak number epk.2 The exterior peak set

Epk(π) of π is defined by

Epk(π) :=

{Lpk(π) ∪ Rpk(π), if |π| 6= 1,

{1}, if |π| = 1,

and the exterior peak number epk(π) of π is defined by

epk(π) := |Epk(π)|.

• The double ascent number dasc. We say that i (where 2 ≤ i ≤ n − 1) is a double

ascent of π ∈ Sn if πi−1 < πi < πi+1. The double ascent number dasc(π) of π is the

number of double ascents of π.

• The right double ascent number rdasc. We say that i (where 2 ≤ i ≤ n) is a right

double ascent of π ∈ Sn if i is a double ascent of π or if i = n and πn−1 < πn. The

right double ascent number rdasc(π) of π is the number of right double ascents of π.

• The exterior double ascent number edasc. We say that i ∈ [n] is an exterior double

ascent of π ∈ Sn if i is a right double ascent of π or if i = 1 and π1 < π2. The2In the present author’s paper [60], the statistic epk is denoted lrpk and the statistic edasc defined below isdenoted lrdasc.

5


exterior double ascent number edasc(π) of π is the number of exterior double ascents

of π.

• The number of biruns br and the number of up-down runs udr. A birun3 of a

permutation is a maximal monotone consecutive subsequence, and the number of

biruns of π is denoted br(π). An up-down run of a permutation π = is either a birun

or π1 when π1 > π2, and the number of up-down runs of π is denoted udr(π). Thus

the up-down runs of π are essentially the biruns of 0π. For example, the biruns of

π = 871542 are 871, 15, and 542, and the up-down runs of π are these biruns along

with 8, so br(π) = 3 and udr(π) = 4.

• Ordered tuples of permutation statistics, such as (pk, des), (lpk, des), and so on.

The distribution of a permutation statistic st over a set S ⊆ Sn is the multiset { st(π) : π ∈ S },

and one of the primary goals of permutation enumeration is to study the distributions of

permutation statistics over Sn and interesting subsets of Sn. Distributions of statistics can be

encoded by polynomials; for example, we already know from (1) that [n]q!—a polynomial in

q—encodes the distribution of the inversion number over Sn. Furthermore, the nth Eulerian

polynomial An(t) defined by An(t) :=∑

π∈Sntdes(π)+1 for n ≥ 1 and by A0(t) := 1 encodes

the distribution of the descent number over Sn. The exponential generating function∞∑n=0

An(t)xn

n!=

1− t1− te(1−t)x .

for Eulerian polynomials is well known, and gives a complete description of the distribution

of the descent number over Sn for all n. The Eulerian polynomials have a rich history and

appear in many contexts in combinatorics outside of permutation enumeration; see [43] for a

detailed exposition.3Biruns are more commonly called “alternating runs”, but since the term “alternating run” is used for adifferent concept in this dissertation (see Section 1.4), we use the term “birun” which was suggested by Stanley[53, Section 4].

6


A permutation statistic st is called a descent statistic if it depends only on the descent

composition, that is, if Comp(π) = Comp(σ) implies st(π) = st(σ) for any two permutations

π and σ. Equivalently, st is a descent statistic if it depends only on the descent set and

length of a permutation. Aside from the inversion number, all of the permutation statistics

mentioned thus far are descent statistics.

Before continuing, we give two lemmas that will help us understand some of the above

statistics. The first lemma characterizes several statistics in terms of “increasing run statistics”.

Let us call an increasing run short if it has length 1, and long if it has length at least 2. The

initial run of a permutation refers to its first increasing run, whereas the final run refers to

its last increasing run. For example, the initial run of 21479536 is 2 and its final run is 36.

(If a permutation has only one increasing run, then it is considered to be both an initial run

and a final run.)

We introduce the following statistics based on increasing runs: lr, lir, lfr, sir, and sfr. Let

lr(π) be the number of long runs of π, let lir(π) be 1 if the initial run of π is long and 0

otherwise, and let lfr(π) be 1 if the final run of π is long and 0 otherwise. Also, if |π| ≥ 1,

let sir(π) := 1− lir(π) and sfr(π) := 1− lfr(π). (By convention, if |π| = 0, then all of these

statistics are equal to zero.)

Lemma 1.3. Let π ∈ Sn with n ≥ 1. Then:

(a) pk(π) = lr(π)− lfr(π)

(b) val(π) = lr(π)− lir(π)

(c) lpk(π) =

{lr(π) + sir(π)− lfr(π), if n ≥ 2,

0, otherwise.

(d) rpk(π) = lr(π)

(e) epk(π) = val(π) + 1

(f) br(π) = pk(π) + val(π) + 1

7


Proof. Part (a) follows from the fact that every non-final long run ends in a peak, and

every peak is at the end of a non-final long run. The same is true for valleys and non-initial

long runs, and for right peaks and long runs, thus implying (b) and (d). Next,

lpk(π) =

{pk(π) + sir(π), if n ≥ 2,

0, otherwise,

which together with (a) proves (c). Furthermore,

epk(π) = rpk(π) + sir(π) = lr(π) + 1− lir(π) = val(π) + 1

proves (e). Finally, part (f) follows from the observation that every peak and valley is at the

end of a birun, and this accounts for every birun except the final birun. �

Our second lemma reveals a close connection between the udr statistic and the lpk and

val statistics.

Lemma 1.4. Let π ∈ Sn with n ≥ 1. Then:

(a) udr(π) = lpk(π) + val(π) + 1

(b) lpk(π) = budr(π)/2c

(c) val(π) = b(udr(π)− 1)/2c

(d) If n ≥ 2 and n − 1 is a descent of π, then lpk(π) = val(π) + 1. Otherwise,

lpk(π) = val(π).

Proof. Every up-down run except the final one ends with either a left peak or a valley,

and in fact these up-down runs alternate between ending with a left peak and ending with a

valley, beginning with a left peak. For example, if udr(π) = 5, then the first up-down run

ends with a left peak, the second ends with a valley, the third ends with a left peak, and the

fourth ends with a valley. It is clear that this accounts for every left peak and every valley,

which proves (a). Now, note that either lpk(π) = val(π) + 1 or lpk(π) = val(π); this depends

completely on whether the penultimate up-down run ends with a left peak or a valley, which8


is determined by whether the final up-down run is increasing or decreasing (i.e., whether the

final run is long or short); this proves (d). Finally, (b) and (c) follow from (a) and (d). �

Lemma 1.4 shows that not only does (lpk, val) determines udr, but udr also determines

(lpk, val). In other words, udr and (lpk, val) are equivalent permutation statistics in the sense

that will be formally defined in Section 5.2.

We give two key remarks before continuing. First, the definitions and properties of

descents, increasing runs, descent compositions, and descent statistics extend naturally to

words on any totally ordered alphabet such as [n] or P (the positive integers) if we replace

the strict inequality < with the weak inequality ≤, which reflects the fact that increasing

runs are allowed to be weakly increasing in this setting. For example, i is a peak of the

word w = w1w2 · · ·wn if wi−1 ≤ wi > wi+1. Given an alphabet A, we let A∗ denote the set of

words on A.4

Finally, recall that by definition, two permutations (or words) with the same descent

composition must have the same value of st if st is a descent statistic. Hence, we shall use

the notation st(L) to indicate the value of a descent statistic st on any permutation (or word)

with descent composition L.

1.3. Possible values of some descent statistics

Later on, it will be useful to determine all possible values that certain descent statistics

can achieve. It is clear that for π ∈ Sn and n ≥ 1, we have 0 ≤ des(π) ≤ n− 1 and des(π)

can attain any value in this range for some π ∈ Sn. It is also easy to check that the possible

values of maj(π) for π ∈ Sn range from 0 to(n2

), and that all of these values are attainable.

Finding such bounds for other descent statistics requires more work. Here, we determine all

possible values for the (pk, des), (lpk, des), and (udr, des) statistics.4In Section 2.5, we briefly use the notation V ∗ to indicate the dual of a vector space V , but this should causeno confusion.

9


Proposition 1.5 (Possible values of (pk, des)).

(a) For any permutation π ∈ Sn with n ≥ 1, we have 0 ≤ pk(π) ≤ b(n− 1)/2c. In

addition, pk(π) ≤ des(π) ≤ n− pk(π)− 1.

(b) If n ≥ 1, 0 ≤ j ≤ b(n− 1)/2c, and j ≤ k ≤ n− j − 1, then there exists π ∈ Sn with

pk(π) = j and des(π) = k.

Proof. Fix n ≥ 1. Recall from Lemma 1.3 (a) that pk(π) is equal to the number of

non-final long runs of π. It is clear that the number of non-final long runs of an n-permutation

is between 0 and b(n− 1)/2c. Every peak is a descent, so pk(π) ≤ des(π). For each peak

i, note that i − 1 ∈ [n − 1] is not a descent, so that pk(π) ≤ n − 1 − des(π) and therefore

des(π) ≤ n− pk(π)− 1. This proves (a).

To prove (b), it suffices to show that if n ≥ 1, 0 ≤ j ≤ b(n− 1)/2c, and j ≤ k ≤ n− j− 1

then there exists a composition of n with j non-final long parts (i.e., parts of size at least 2)

and k + 1 total parts. Such a composition is (2j, 1k−j, n− k − j). Hence, (b) is proved. �

Proposition 1.6 (Possible values of (lpk, des)).

(a) For any permutation π ∈ Sn with n ≥ 1, we have 0 ≤ lpk(π) ≤ bn/2c. In addition,

if lpk(π) = 0, then des(π) = 0; otherwise, lpk(π) ≤ des(π) ≤ n− lpk(π).

(b) If n ≥ 1, 1 ≤ j ≤ bn/2c, and j ≤ k ≤ n − j, then there exists π ∈ Sn with

lpk(π) = j and des(π) = k. In addition, for any n ≥ 1, there exists π ∈ Sn with

lpk(π) = des(π) = 0.

Proof. If lpk(π) = 0, then π is an increasing permutation, so des(π) = 0. The other

inequalities of part (a) follow from applying Proposition 1.5 (a) to the permutation 0π.5

Now, fix n ≥ 2. (The case n = 1 is obvious.) The increasing permutation with descent

composition (n) has no left peaks and no descents. Suppose that 1 ≤ j ≤ bn/2c and5Technically 0π is not an element of Sn, but Proposition 1.5 (a) holds for permutations on any set of integers.

10


j ≤ k ≤ n− j. To complete the proof of (b), we show that there exists a composition L of n

with exactly k + 1 parts such that lpk(L) = lr(L) + sir(L)− lfr(L) = j. Such a composition

is (1k−j+1, 2j−1, n− k − j + 1). This completes the proof of (b). �

Let asc(π) denote the number of ascents of π. It is clear that des(π) = n− 1− asc(π).

Proposition 1.7 (Possible values of (udr, des)).

(a) For any permutation π ∈ Sn with n ≥ 1, we have 1 ≤ udr(π) ≤ n. In addition, if

udr(π) = 1, then des(π) = 0; otherwise, budr(π)/2c ≤ des(π) ≤ n− dudr(π)/2e.

(b) If n ≥ 1, 2 ≤ j ≤ n, and bj/2c ≤ k ≤ n − dj/2e, then there exists π ∈ Sn with

lpk(π) = j and des(π) = k. In addition, for any n ≥ 1, there exists π ∈ Sn with

udr(π) = 1 and des(π) = 0.

Proof. It is clear that every nonempty permutation has at least one up-down run, and

every up-down run of a permutation ends with a different letter, so 1 ≤ udr(π) ≤ n. The

beginning of the 2ith up-down run of π is always a descent of π, so des(π) ≥ budr(π)/2c.

The beginning of the (2i− 1)th up-down run of π is an ascent of π for i ≥ 2, so the number

of ascents of π is at least b(udr(π)− 1)/2c = dudr(π)/2e − 1. Thus

des(π) = n− 1− asc(π) ≤ n− 1− (dudr(π)/2e − 1) = n− dudr(π)/2e,

completing the proof of (a). Now, fix n ≥ 2. (The case n = 1 is obvious.) The increasing

permutation with descent composition (n) has only one up-down run and no descents. Suppose

that 1 ≤ j ≤ n and bj/2c ≤ k ≤ n−dj/2e. To complete the proof of (b), we show that there

exists a composition L of n with exactly k+ 1 parts such that udr(L) = lpk(L) + val(L) + 1 =

2 sir(L) + 2 lr(L)− lfr(L) = j. For this, we consider three cases:

• If j = 2, then we can take (n− k, 1k).

• If j > 2 and j is even, then we can take (1, n− j/2− k + 2, 2j/2−2, 1k−j/2+1).

• If j is odd, then we can take (1k+1−(j−1)/2, 2(j−3)/2, n− (j + 1)/2− k + 2).11


This completes the proof of (b). �

1.4. Alternating permutations, descents, and runs

Following Stanley [54], we say that π is an alternating permutation if π1 > π2 < π3 >

π4 < · · · . If instead π1 < π2 > π3 < π4 > · · · , then we say that π is reverse-alternating. It is

well known that the number of alternating n-permutations is the nth Euler number En defined

by∑∞

n=0 Enxn/n! = secx+tanx. Since alternating n-permutations are in clear bijection with

reverse-alternating n-permutations via complementation—that is, π is alternating implies πc is

reverse-alternating and vice versa—the Euler numbers count reverse-alternating permutations

as well.

In [7], Chebikin introduced a variant of the notion of descents which is closely related to

alternating permutations and the Euler numbers: i ∈ [n− 1] is called an alternating descent

of π if i is odd and πi > πi+1 or if i is even and πi < πi+1. We define an alternating run

of π to be a maximal consecutive subsequence of π containing no alternating descents. For

example, the alternating runs of the permutation 3421675 are 342, 1, and 675. An alternating

run starting in an odd position is a reverse-alternating permutation and an alternating run

starting in an even position is an alternating permutation.

The notions of alternating descents and alternating runs give rise to an “alternating

analogue” for nearly every concept introduced thus far relating to descents. For example, the

alternating descent set Altdes, the alternating descent number altdes, and the alternating

descent composition are all defined in the obvious way. The distribution of the alternating

descent number altdes over Sn is given by the nth alternating Eulerian polynomial defined

by

An(t) :=∑π∈Sn

taltdes(π)+1

12


for n ≥ 1 and by A0(t) := 1; this is the alternating analogue of the nth Eulerian polynomial.

The exponential generating function for the alternating Eulerian polynomials is∞∑n=0

An(t)xn

n!=

1− t1− t(sec((1− t)x) + tan((1− t)x))

,

which is the exponential generating function for the ordinary Eulerian polynomials with the

exponential function ex replaced by secx+ tanx.

Here we give alternating analogues of Lemmas 1.1 (a) and 1.2 (a). If L = (L1, . . . , Lk) is

a composition of n, then we write(nL

)Efor(nL

)EL1 · · ·ELk

.

Lemma 1.8. Let L be a composition of n. Then the number of n-permutations with

alternating descent composition K ≤ L—or equivalently, with alternating descent set contained

in Des(L)—is(nL

)E.

Proof. Let L = (L1, . . . , Lk) be a composition of n. To create an n-permutation with

alternating descent composition K ≤ L, first choose an ordered partition of [n] with k blocks,

where the ith block Bi has size Li for each 1 ≤ i ≤ k; there are(nL

)such partitions to choose

from. Let bi = bi−1 + Li−1 for i ≥ 2 and let b1 = 1. Now, for each 1 ≤ i ≤ k, arrange the

letters of Bi into a permutation wi, where wi is alternating if bi is even and reverse-alternating

if bi is odd; there are EL1 · · ·ELkchoices for these permutations w1, . . . wk.

Let π be the n-permutation obtained by concatenating w1 · · ·wk. Since there are no

alternating descents within each wi, it follows that π has alternating descent composition

K ≤ L. Moreover, it is clear that every n-permutation with alternating descent composition

K ≤ L can be obtained in this way, so there are exactly(nL

)E

=(nL

)EL1 · · ·ELk

such

permutations. �

Lemma 1.9. Let L be a composition of n. Then the number β(L) of n-permutations with

alternating descent composition L—or equivalently, with alternating descent set Des(L)—is

13


given by the formula

β(L) =∑K≤L

(−1)l(L)−l(K)

(n

K

)E

.

Proof. Follows immediately from Lemma 1.8 and inclusion-exclusion. �

14

CHAPTER 2

Noncommutative symmetric functions

2.1. Basic definitions

We now introduce relevant aspects of the theory of noncommutative symmetric functions,

which were first studied per se by Gelfand, et al. [18] in 1995 but have appeared implicitly in

Ira Gessel’s Ph.D. dissertation [24].

Throughout this chapter, fix a field F of characteristic zero. (In our applications, we can

take F to be Q.) Let F 〈〈X1, X2, . . . 〉〉 be the F -algebra of formal power series in countably

many noncommuting variables X1, X2, . . . . Consider the elements

hn :=∑

i1≤···≤in

Xi1Xi2 · · ·Xin

of F 〈〈X1, X2, . . . 〉〉, with h0 := 1, which are noncommutative versions of the complete

symmetric functions hn. Note that hn is the noncommutative generating function for weakly

increasing words of length n on the alphabet P. For example, the weakly increasing word

13449 is encoded by X1X3X24X9, which appears as a term in h5. Given a composition

L = (L1, . . . , Lk), we let

hL := hL1 · · ·hLk. (3)

Then

hL =∑i1,...,in

Xi1Xi2 · · ·Xin

where the sum is over all i1, . . . , in satisfying

i1 ≤ · · · ≤ iL1︸︷︷︸L1

, iL1+1 ≤ · · · ≤ iL1+L2︸︷︷︸L2

, . . . , iL1+···+Lk−1+1 ≤ · · · ≤ in︸︷︷︸Lk

,

15

CHAPTER 2. NONCOMMUTATIVE SYMMETRIC FUNCTIONS

so hL is the noncommutative generating function for words in P whose descent set is

contained in Des(L), or equivalently, whose descent composition K satisfies K ≤ L in the

reverse refinement ordering.

Let Symn denote the vector space spanned by {hL}L�n, and let Sym :=⊕∞

n=0 Symn.

Then Sym is a graded F -algebra called the algebra of noncommutative symmetric functions

with coefficients in F , a subalgebra of F 〈〈X1, X2, . . . 〉〉. The elements of Sym are called

noncommutative symmetric functions.1

For a composition L = (L1, . . . , Lk), we define

rL :=∑i1,...in

Xi1Xi2 · · ·Xin

where the sum is over all i1, . . . , in satisfying

i1 ≤ · · · ≤ iL1︸︷︷︸L1

> iL1+1 ≤ · · · ≤ iL1+L2︸︷︷︸L2

> · · · > iL1+···+Lk−1+1 ≤ · · · ≤ in︸︷︷︸Lk

.

Then, rL is the noncommutative generating function for words on the alphabet P with descent

composition L.

Note that

hL =∑K≤L

rK , (4)

so by inclusion-exclusion,

rL =∑K≤L

(−1)l(L)−l(K)hK . (5)

Hence the rL are noncommutative symmetric functions, and are in fact noncommutative

versions of the ribbon skew Schur functions rL.

Since rL and rM have no terms in common for L 6= M , it is clear that {rL}L�n is linearly

independent. From (4), we see that {rL}L�n spans Symn, so {rL}L�n is a basis for Symn.1In practice, we will oftentimes be working in the completion of Sym, which allows for infinite sums ofnoncommutative symmetric functions of unbounded degree. By an abuse of notation, we also use Sym todenote the completion of Sym.

16


Because {hL}L�n spans Symn and has the same cardinality as {rL}L�n, we conclude that

{hL}L�n is also a basis for Symn.

Let us also consider the noncommutative generating function

en :=∑

i1>···>in

Xi1Xi2 · · ·Xin

for decreasing words of length n on the alphabet P. Then en is a noncommutative version of

the elementary symmetric function en, and en ∈ Symn since en = r(1n).

Let

h(x) :=∞∑n=0

hnxn

be the generating function for the noncommutative complete symmetric functions hn, where

x commutes with all of the variables Xi, and let

e(x) :=∞∑n=0

enxn

be the generating function for the en. Then we have

e(x) = h(−x)−1,

a consequence of the infinite product formulas

h(x) = (1−X1x)−1(1−X2x)−1 · · · and e(x) = · · · (1 +X2x)(1 +X1x)

(cf. [24, p. 38] and [18, Section 7.3]).

Although we won’t need to use this fact in this dissertation, it is worth noting that for a

composition L = (L1, . . . , Lk) of n, we can define

eL := eL1eL2 · · · eLk

and {eL}L|=n is a third basis for Symn. This can be proven using a noncommutative analogue

of the ω involution for ordinary symmetric functions (see [56, Section 7.6]).

17


2.2. Homomorphisms on Sym

Many results in the next two chapters are obtained by applying certain homomorphisms

to various identities involving noncommutative symmetric functions. The simplest of these

homomorphisms is the map Φ: Sym→ F [[x]] defined by Φ(hn) = xn/n!. We now give an

alternating analogue and a q-analogue of Φ. Define the homomorphism Φ : Sym→ F [[x]] by

Φ(hn) = Enxn/n! and define the homomorphism Φq : Sym→ F [[q, x]] by Φq(hn) = xn/[n]q!.

Then if L is a composition of n, we have

Φ(hL) =xL1

L1!· · · x

Lk

Lk!=

(n

L

)xn

n!,

Φ(hL) = EL1

xL1

L1!· · ·ELk

xLk

Lk!=

(n

L

)E

xn

n!,

and

Φq(hL) =xL1

[L1]q!· · · x

Lk

[Lk]q!=

(n

L

)q

xn

[n]q!.

For our work in the next two chapters, we also need to determine the effect of these

homomorphisms on rL, h(1) =∑∞

n=0 hn, and e(1) =∑∞

n=0 en. Recall that β(L) is the number

of n-permutations with descent composition L, β(L) is the number of n-permutations with

alternating descent composition L, and βq(L) is the polynomial counting n-permutations

with descent composition L by inversion number.

Lemma 2.1.

(a) Let L be a composition of n. Then Φ(rL) = β(L)xn/n!.

(b) Φ(h(1)) = ex.

(c) Φ(e(1)) = ex.

Proof. Part (a):

Φ(rL) = Φ(∑K≤L

(−1)l(L)−l(K)hK

), by (5)

18


=∑K≤L

(−1)l(L)−l(K)Φ(hK)

=∑K≤L

(−1)l(L)−l(K)

(n

K

)xn

n!

= β(L)xn

n!, by (2).

Part (b):

Φ(h(1)) = Φ( ∞∑n=0

hn

)=∞∑n=0

Φ(hn)

=∞∑n=0

xn

n!

= ex.

Part (c):

Φ(e(1)) = Φ( ∞∑n=0

en

)=∞∑n=0

Φ(r(1n))

=∞∑n=0

xn

n!, by part (a)

= ex. �

We omit the proofs of the analogous results for Φ and Φq since they proceed in exactly

the same way.

Lemma 2.2.

(a) Let L be a composition of n. Then Φ(rL) = β(L)xn/n!.

19


(b) Φ(h(1)) = sec(x) + tan(x).

(c) Φ(e(1)) = sec(x) + tan(x).

Consider the q-exponential function

expq(x) :=∞∑n=0

xn

[n]q!

and its variant

Expq(x) :=∞∑n=0

q(n2) xn

[n]q!,

both q-analogues of the classical exponential function ex.

Lemma 2.3.

(a) Let L be a composition of n. Then Φq(rL) = βq(L)xn/[n]q!.

(b) Φq(h(1)) = expq(x).

(c) Φq(e(1)) = Expq(x).

The homomorphisms Φ, Φ, and Φq give us a general principle that whenever we have an

exponential generating function that counts permutations with a restriction on increasing

run lengths, there is an analogous exponential generating function—obtained by replacing

xn/n! by Enxn/n!—for counting permutations with the same restriction on alternating run

lengths, as well as an analogous q-exponential generating function—obtained by replacing

xn/n! by xn/[n]q!—for counting permutations with the same restriction on increasing run

lengths but also keeping track of the inversion number.

For example, take the exponential generating function[∞∑n=0

( xmn

(mn)!− xmn+1

(mn+ 1)!

)]−1

(6)

for permutations with all increasing runs having length less than m, a classical result of David

and Barton [9]. By Lemma 2.1 (a), David and Barton’s formula (6) follows from applying the

20


homomorphism Φ to the equation in the following lemma, which we will prove using Gessel’s

run theorem in Section 3.2.

Lemma 2.4. Let m be a positive integer. Then∑L

rL =( ∞∑n=0

(hmn − hmn+1))−1

(7)

where the sum on the left is over all compositions L with all parts less than m.

If we were to apply Φ to (7) instead, then by Lemma 2.2 (a), we would obtain the formula[∞∑n=0

(Emn

xmn

(mn)!− Emn+1

xmn+1

(mn+ 1)!

)]−1

(8)

for the exponential generating function counting permutations with all alternating runs

having length less than m. And by Lemma 2.3 (a), applying Φq to (7) yields the formula[∞∑n=0

( xmn

[mn]q!− xmn+1

[mn+ 1]q!

)]−1

for the q-exponential generating function counting permutations with all increasing runs

having length less than m, where the variable q is keeping track of the inversion number

(cf. [12, Section 4.2]).

In Chapter 3, we will relate the formula (8) to the enumeration of permutations with all

peaks odd and all valleys even. Also see the present author’s paper [23] for other applications

of the homomorphism Φ which are not present in this dissertation, including formulas of

Carlitz [6], Chebikin [7], and Remmel [45].

2.3. Several noncommutative symmetric function formulas

In this section, we give several noncommutative symmetric function formulas relating to

descent statistics which will be used in proving some of our main results in the upcoming

chapters. All of these formulas can be proven using the generalized run theorem (see Section

21


3.3) and making appropriate substitutions, but here we provide elementary combinatorial

proofs for these formulas.

Lemma 2.5.

(1− te(yx)h(x))−1 =

1

1− t+∞∑n=1

∑L�n

tpk(L)+1(y + t)des(L)−pk(L)(1 + yt)n−pk(L)−des(L)−1(1 + y)2 pk(L)+1

(1− t)n+1xnrL

Proof. Let P = {1, 2, 3, . . . } denote the set of positive integers decorated with underlines,

endowed with the usual total ordering of P. Let us say that a word w on the alphabet P∪P∪{|}

(that is, the positive integers, underlined positive integers, and a vertical bar) is a peak word

if w can be written as a sequence of subwords of the form w1w2| where w1 is a (possibly

empty) strictly decreasing word containing only letters from P and w2 is a (possibly empty)

weakly increasing word containing only letters from P. For example,

864211|457|931||12338|56|||942788| (9)

is a peak word. It is clear that the left-hand side of the given equation counts peak words

where t is weighting the number of bars, y is weighting the number of underlined letters, and

x is weighting the length of the underlying word in P∗. We want to show that the right-hand

side also counts peak words with the same weights.

Let us say that a peak word is minimal if it is impossible to remove bars from it to yield a

peak word. Given a word in P∗, there is a unique minimal peak word corresponding to every

possible choice of underlines. Indeed, if w is a word in P∗ with a given choice of underlines

(that is, if w is a word on the alphabet P∪ P), then a minimal peak word corresponding to w

must have no bar at the beginning and a bar at the end, and whether or not there needs to

be a bar between two letters a and b is completely determined by whether a > b, whether

a is underlined, and whether b is underlined. Moreover, adding bars to a peak word yields

22


another peak word, so every peak word can be obtained from a unique minimal peak word

by adding bars. For example, the minimal peak word corresponding to

8642114579311233856942788

is

864211457|931|12338|56|942788|,

which is the unique minimal peak word from which we can obtain (9) as they share the same

underlying word in P∗ and choice of underlines.

We show that

t(t+ yt)pk(L)(1 + y)pk(L)+1(y + t)des(L)−pk(L)(1 + yt)n−des(L)−pk(L)−1xnrL (10)

counts nonempty minimal peak words with descent composition L � n. Every term in rL

corresponds to a word in P∗ with descent composition L, and we give it a choice of underlines

and insert necessary bars. As our working example, take the word 11375438876544579756673.

(1) There must be a bar at the end, hence the initial factor t:

11375438876544579756673|.

(2) For each letter corresponding to a peak, we choose whether or not to underline it. If

we do underline it, then we insert a bar immediately before it; otherwise, we insert a

bar immediately after it. This corresponds to the (t+ yt)pk(L) factor. For example,

we may have

113|754388|7654457|97566|73|.

(3) The above step divides our word into pk(L) + 1 segments, separated by bars. Take

the left-most smallest letter of each segment and choose whether or not to underline

23


it; this gives the (1 + y)pk(L)+1 factor. For example, we may have

113|754388|7654457|97566|73|.

Note that this step determines whether the left-most smallest letter in each segment

is to be part of the underlined decreasing subword or the non-underlined weakly

increasing subword.

(4) Take each letter corresponding to a descent that is not a peak and choose to either

underline it or to add a bar after it; this gives (y + t)des(L)−pk(L). For example, we

may have

113|754|388|7654457|97|566|73|.

This step eliminates instances of underlined letters separated by non-underlined

letters in the same segment, and it is evident that this gives the minimal peak word

corresponding to our current choice of underlines.

(5) Finally, iterate through every letter that is (a) not the final letter of the word,

(b) not corresponding to a descent, and (c) not followed immediately by a letter

corresponding to a peak, and choose either to do nothing or to underline the next

letter and add a bar in between the two letters; this gives (1 + yt)n−des(L)−pk(L)−1.

For example, we may have

11|3|754|3|88|7654457|97|56|6|73|.

Note that adding these underlines requires the corresponding bars to be placed, so

the result is still a minimal peak word.

Through these steps, we have considered whether to underline each letter in the word, so

in fact (10) accounts for the unique minimal peak word corresponding to each choice of

underlines, and thus counts all minimal peak words with descent composition L � n.

24


Observe that (10) is equal to

tpk(L)+1(y + t)des(L)−pk(L)(1 + yt)n−pk(L)−des(L)−1(1 + y)2 pk(L)+1xnrL,

which appears in the statement of this lemma. Dividing by (1− t)n+1 corresponds to inserting

any number of bars in the n+ 1 possible positions, which allows us to move from nonempty

minimal peak words to all peak words except those that only consist of bars, which are

accounted for by the 1/(1− t) term at the beginning. Hence the lemma is proven. �

Lemma 2.6.

h(x)(1− te(yx)h(x))−1 =

1

1− t+∞∑n=1

∑L�n

tlpk(L)(y + t)des(L)−lpk(L)(1 + yt)n−lpk(L)−des(L)(1 + y)2 lpk(L)

(1− t)n+1xnrL

Proof. Let us say that a word w on the alphabet P ∪ P ∪ {|} is a left peak word if w

begins with a (possibly empty) weakly increasing subword containing only letters from P,

followed by a sequence of subwords of the form |w1w2 where w1 is a (possibly empty) strictly

decreasing word containing only letters from P and w2 is a (possibly empty) weakly increasing

word containing only letters from P. The left-hand side of the given equation counts left

peak words where t is weighting the number of bars, y is weighting the number of underlined

letters, and x is weighting the length of the underlying word in P∗. We want show that the

right-hand side also counts left peak words with the same weights.

Call a left peak word w minimal if it is impossible to remove bars from w to yield a left

peak word. Similar to peak words in the proof of Lemma 2.5, every left peak word can be

obtained from only one minimal left peak word, which is the only minimal left peak word on

those letters with the same choice of underlines. We claim that

(t+ yt)lpk(L)(1 + y)lpk(L)(y + t)des(L)−lpk(L)(1 + yt)n−des(L)−lpk(L)xnrL (11)

25


counts nonempty minimal peak words with descent composition L � n. Every term in rL

corresponds to a word in P∗ with descent composition L, and we give it a choice of underlines

and insert bars in a similar way as in the proof of Lemma 2.5:

(1) For each letter corresponding to a left peak, we choose whether or not to underline

it. If we do underline it, then we insert a bar immediately before it; otherwise, we

insert a bar immediately after it. This corresponds to the (t+ yt)lpk(L) factor.

(2) If the first letter corresponds to a left peak and was underlined, then the bars

inserted in the above step divide our word into lpk(L) segments. In this case, take

the left-most smallest letter of each segment and choose whether or not to underline

it. Otherwise, the bars divide our word into lpk(L) + 1 segments, in which case we

take the left-most smallest letter of each but the first segment and choose whether

or not to underline it. This gives the (1 + y)lpk(L) factor.

(3) Take each letter corresponding to a descent that is not a left peak and choose

to either underline it or to add a bar after it; this gives (y + t)des(L)−lpk(L). As

in the proof of Lemma 2.5, this step eliminates underlined letters separated by

non-underlined letters appearing in the same segment, and gives a minimal left peak

word corresponding to our current choice of underlines.

(4) Finally, iterate through every letter that is (a) not the final letter of the word, (b)

not corresponding a descent, and (c) not followed by a letter corresponding to a

left peak, and choose either to do nothing or to underline the next letter and add a

bar in between the two letters. In addition, if the first letter does not correspond

to a left peak, then choose to either do nothing or to underline the first letter and

prepend a bar. This gives (1 + yt)n−des(L)−lpk(L)−1, and the result is still a minimal

left peak word as the new bars are necessary to accomodate the new underlines.

26


Through these steps, we have considered whether to underline each letter in the word, so (11)

counts every minimal left peak word with descent composition L � n. Dividing by (1− t)n+1

allows us to insert any number of bars in any of the n+ 1 possible positions, thus creating

left peak words from minimal left peak words, and the 1/(1 − t) term accounts for words

containing only bars. �

Lemma 2.7.

(1− t2h(x)e(yx))−1(1 + th(x)) =1

1− t+∞∑n=1

∑L�n

NL

(1− t)(1− t2)nxnrL

where

NL = tudr(L)(1 + y)udr(L)−1(1 + yt2)n−1−des(L)−val(L)(y + t2)des(L)−lpk(L)

× (1 + yt)1−lpk(L)+val(L)(y + t)lpk(L)−val(L).

Proof. Let us say that a word w on the alphabet P ∪ P ∪ {|} is an up-down run word if

w is either:

• A sequence of subwords of the form w1|w2| where w1 is a (possibly empty) weakly

increasing word containing only letters from P and w2 is a (possibly empty) strictly

decreasing word containing only letters from P;

• Or, a sequence of subwords of the form w1|w2| as described above, but ending with

a subword of the form w3|, where w3 is a (possibly empty) weakly increasing word

containing only letters from P.

For example,

12||246678|98|4|321||5|||23| (12)

is an up-down run word. The left-hand side of the given equation counts up-down run words

where, as before, t is weighting the number of bars, y is weighting the number of underlined

letters, and x is weighting the length of the underlying word in P∗. We want show that the

right-hand side also counts up-down run words with the same weights.27


Call an up-down run word w minimal if it is impossible to remove bars from w to yield

an up-down run word. As before, every up-down run word can be obtained from only one

minimal up-down run word, which is the only minimal up-down run word on those letters

with the same choice of underlines. For example, the minimal up-down run word on

12246678984321523

is

12246678|98|4|321||5|23|,

which is the unique minimal up-down run word that (12) can be obtained from. We claim

t(t+ yt)udr(L)−1(1 + yt2)n−1−des(L)−val(L)(y + t2)des(L)−lpk(L)

× (1 + yt)1−lpk(L)+val(L)(y + t)lpk(L)−val(L)xnrL (13)

counts nonempty minimal up-down run words with descent composition L � n. Every term

in rL corresponds to a word in P∗ with descent composition L, and we give it a choice of

underlines and insert the necessary bars. Let us take 85432113444889323344513456 as our

working example.

(1) Every up-down run word must end with a bar, so insert a bar at the end of our word:

85432123444889323344513456|.

This gives the initial factor of t.

(2) For each letter corresponding to a left peak or valley (i.e., each letter that is at

the end of an up-down run other than the last one), we choose whether or not to

underline it. For a left peak, if we do underline it, then we insert a bar immediately

before it; otherwise, insert a bar immediately after it. For a valley, if we do underline

it, then we insert a bar immediately after it; otherwise, insert a bar immediately

28


before it. This gives the (t+ yt)udr(L)−1 factor. For example, we may have

|85432|12344488|932|33445|1|3456|.

(3) For each letter corresponding to a descent that is not a left peak (i.e., each letter

corresponding to a descent and is not the final letter of an up-down run), choose

whether or not to underline it. If we do not underline the letter, then prepend and

append a bar to it. This gives the (y + t2)des(L)−lpk(L) factor. For example, we may

have

|854|3|2|12344488|932|33445|1|3456|.

This step eliminates instances of non-underlined letters appearing in the same segment

as an underlined letter, and by adding the bars, we have a minimal up-down run

word corresponding to our current choice of underlines.

(4) For each letter corresponding to an ascent that is not a valley (i.e., each letter

corresponding to an ascent and is not the final letter of an up-down run), choose

whether or not to underline it. If we underline the letter, then also prepend and

append a bar to it. This gives the (1 + yt2)n−1−des(L)−val(L) factor. For example, we

may have

|854|3|2|12|3||4|4488|932|33|4|45|1|3456|.

Note that adding the bars is necessary so that the result is a minimal up-down run

word.

(5) The only remaining letter of our word that still requires consideration is the final

letter, so the last step is to choose whether or not to underline it. If the word ends

with an increasing run of length 1 (which is equivalent to lpk(L) − val(L) = 1 by

Lemma 1.4)2 and we do not underline the final letter, then prepend a bar to it. If2Although Lemma 1.4 was stated for permutations, it also holds for words.

29


the word ends with an increasing run of length at least 2 (which is equivalent to

lpk(L)− val(L) = 0 by Lemma 1.4) and we underline the final letter, then prepend

a bar to it. This gives (1 + yt)1−lpk(L)+val(L)(y + t)lpk(L)−val(L). For example, we may

have

|854|3|2|12|3||4|4488|932|33|4|45|1|345|6|.

Again, we have a minimal up-down run word.

We have chosen whether or not to underline each letter in the word, so (13) counts every

minimal up-down run word with descent composition L � n. Dividing by (1− t2)n allows us

to insert bars in multiples of two at the beginning of the word or between any two letters;

adding them in multiples of two is necessary for the result to remain an up-down word.

However, any number of bars can be added at the end, hence dividing by 1− t as well. This

accounts for all up-down words other than those only consisting of bars, which are accounted

for by the 1/(1− t) term. �

Corollary 2.8.

(1− t2h(x)e(x))−1(1 + th(x)) =1

1− t+∞∑n=1

∑L�n

2udr(L)−1tudr(L)(1 + t2)n−udr(L)

(1− t)2(1− t2)n−1xnrL

Proof. This follows easily from setting y = 1 in Lemma 2.7 and simplifying using

udr(L) = lpk(L) + val(L) + 1 (Lemma 1.4). �

2.4. The bialgebra structure of Sym

Later, in Chapter 5, we shall need to define a “bialgebra structure” on noncommutative

symmetric functions. In order to define a bialgebra, we require the following equivalent

definition of an algebra. Let R be a commutative ring. An R-algebra A is an R-module with

an R-linear map µ : A⊗ A→ A such that the following diagram commutes:

30


A⊗ A⊗ A id⊗µ−−−→ A⊗ A

µ⊗id

y yµA⊗ A −−−→

µA

The map µ is called a multiplication.3

The notion dual to an algebra is a coalgebra, defined as follows. An R-coalgebra C is an

R-module with an R-linear map ∆: C → C ⊗ C such that the following diagram commutes:

C ⊗ C ⊗ C id⊗∆←−−− C ⊗ C

∆⊗id

x x∆

C ⊗ C ←−−−∆

C

Observe that this diagram is essentially the diagram in the definition of an algebra, but with

arrows reversed. The map ∆ is called a comultiplication.4

If an R-module A is simultaneously an R-algebra and an R-coalgebra such that its

comultiplication map is an R-algebra homomorphism, then we call A an R-bialgebra.

The algebra Sym can be given a coalgebra structure by defining the comultiplication

∆: Sym→ Sym by

∆hn =n∑i=0

hi ⊗ hn−i (14)

and extending by the rule

∆(fg) = (∆f)(∆g).

Since the comultiplication ∆ is an algebra homomorphism, Sym is a bialgebra.3The multiplication map µ satisfies µ(a⊗ b) = ab under the original definition of an algebra; from this, it isclear why µ is called “multiplication”.4Typically, the definition of an algebra requires an additional linear map called a “unit” which satisfies acertain commutative diagram, and the definition of a coalgebra requires the dual concept of a “counit”, butthese will not be necessary for our work.

31


2.5. Quasisymmetric functions

A formal power series f ∈ F [[x1, x2, . . . ]] of bounded degree in countably many commuting

variables x1, x2, . . . is called a quasisymmetric function if for any positive integers a1, a2, . . . , ak,

if i1 < i2 < · · · < ik and j1 < j2 < · · · < jk, then

[xa1i1 xa2i2· · ·xakik ] f = [xa1j1 x

a2j2· · ·xakjk ] f.

It is clear that every symmetric function is quasisymmetric, but not every quasisymmetric

function is symmetric. For example,∑

i<j<k x2ixjxk is quasisymmetric, but it is not symmetric

because x21x2x3 appears as a term yet x1x

22x3 does not.

Let QSymn be the set of quasisymmetric functions homogeneous of degree n, which is

clearly a vector space. For a composition L = (L1, L2, . . . , Lk), the monomial quasisymmetric

function ML is defined by

ML :=∑

i1<i2<···<ik

xL1i1xL2i2. . . xLk

ik

It is clear that {ML}L�n is a basis for QSymn, so for n ≥ 1, QSymn has dimension 2n−1, the

number of compositions of n.

Another important basis for QSymn (and the most important basis for our purposes) is

the basis of fundamental quasisymmetric functions {FL}L�n given by

FL :=∑

i1≤i2≤···≤inij<ij+1 if j∈Des(L)

xi1xi2 · · ·xin .

It is easy to see that

FL =∑

Des(K)⊇Des(L)|K|=|L|

MK ,

so by inclusion-exclusion, MK can be expressed as a linear combination of the FL. It follows

that {FL}L�n spans QSymn, so this set must be a basis for QSymn since it has the correct

number of elements.

32


The product of two quasisymmetric functions is quasisymmetric, with the product formula

for the fundamental basis given by the following theorem, which may be proved using P -

partitions; see [56, Exercise 7.93]. Here, the notions of “permutations” and “shuffles” are

those defined in Section 5.1.

Theorem 2.9. Let cLJ,K be the number of permutations with descent composition L among

the shuffles of a permutation π with descent composition J and a permutation σ (disjoint

from π) with descent composition K. Then

FJFK =∑L

cLJ,KFL.

If f ∈ QSymm and g ∈ QSymn, then fg ∈ QSymm+n. Thus QSym :=⊕∞

n=0 QSymn

is a graded F -algebra called the algebra of quasisymmetric functions with coefficients in

F , a subalgebra of F [[x1, x2, . . . ]]. Motivated by Richard Stanley’s theory of P -partitions,

Ira Gessel introduced quasisymmetric functions in [19] and developed the basic algebraic

properties of QSym. Further properties of QSym and connections with many topics of study

in combinatorics and algebra were developed in the subsequent decades. Basic references

include [56, Section 7.19], [30, Section 5], and [37].

Suppose now that R is a field and that V =⊕

n≥0 Vn is a graded R-vector space of

finite type, that is, each component Vn is finite-dimensional. Let V o denote the graded

dual V o :=⊕

n≥0 V∗n , which is contained inside the dual space V ∗ of V . We say that a

linear map φ : V → W is graded if, for every n ≥ 0, φ(Vn) is contained inside the nth

homogeneous component of W . Every graded linear map φ : V → W induces a graded linear

map φo : W o → V o given by

φo(f)(v) = f(φ(v))

for f ∈ W o and v ∈ V . In particular, if A is a graded R-algebra—meaning that its vector

space and multiplication are graded—and is of finite type, then by reversing the arrows in

33


the commutative diagram, we see that Ao has the structure of a graded R-coalgebra. In fact,

if A has basis {ai} with structure constants {cij,k}, i.e.,

ajak =∑i

cij,kai,

then the {cij,k} are also the structure constants for the comultiplication of the dual basis {fi}

in Ao:

∆(fi) =∑j,k

cij,kfj ⊗ fk.

Similarly, the graded dual of a graded R-coalgebra is a graded R-algebra, with the same

correspondence of structure constants. If φ is an R-algebra homomorphism, then φo is an

R-coalgebra homomorphism, and vice versa.

We now show that the graded dual of the algebra QSym is the coalgebra Sym; cf. [18,

Theorem 6.1] or [30, Subsection 5.3]. We may extend the definition of hL to weak compositions

L by (3), so that if L is a weak composition then hL = hL′ where L′ is the composition

obtained from L by removing all zero parts. For two weak compositions J = (J1, J2, . . . , Jk)

and K = (K1, K2, . . . , Kk) with the same number of parts, let J + K denote the weak

composition (J1 +K1, J2 +K2, . . . , Jk +Kk) obtained by summing the entries of J and K

componentwise.

Lemma 2.10. Let L be a composition. Then ∆hL =∑

J,K hJ ⊗hK , where the sum is over

all pairs of weak compositions J and K with the same number of parts such that J +K = L.

Proof. This follows easily from the fact that ∆h(L1,...,Lm) = ∆hL1 · · ·∆hLm together

with (14). �

Theorem 2.11. The graded dual of the algebra QSym of quasisymmetric functions is

isomorphic to the coalgebra Sym of noncommutative symmetric functions. In particular,

34


the monomial basis {ML} of QSym is dual to the complete basis {hL} of Sym and the

fundamental basis {FL} of QSym is dual to the ribbon basis {rL} of Sym.

Proof. We first consider the product of two monomial quasisymmetric functions. Define

coefficients bLJ,K by

MJMK =∑L

bLJ,KML. (15)

It is easy to see that bLJ,K is the number of pairs of weak compositions (J ′, K ′) with the same

number of parts such that J ′ is obtained from J by inserting zeros, K ′ is obtained from K

by inserting zeros, and J ′ +K ′ = L.

Lemma 2.10 implies that

∆hL =∑J,K

bLJ,KhJ ⊗ hK ,

where the coefficients bLJ,K are the same as those in equation (15). Thus {ML}L�n and {hL}L�n

are dual bases for QSymn and Symn.

We may define a pairing between QSym and Sym by

〈MK ,hL〉 = δK,L =

{1, if K = L,

0, otherwise.

Then

〈FK , rL〉 =

⟨ ∑Des(I)⊇Des(K)

MI ,∑

Des(J)⊆Des(L)

(−1)l(L)−l(J)hJ

⟩

=∑

Des(K)⊆Des(J)⊆Des(L)

(−1)l(L)−l(J) = δK,L,

and this implies that {FL} and {rL} are dual bases. �

We note that both Sym and QSym are Hopf algebras (see [30] for a definition), and

QSym plays an important role as the terminal object in the category of combinatorial Hopf

algebras in the sense of Aguiar–Bergeron–Sottile [3]. The duality between Sym and QSym

35


given above extends to a Hopf algebra duality, but we will not use antipodes or the coalgebra

structure of QSym in this dissertation.

36

CHAPTER 3

The run theorem and its applications

3.1. Introduction

In his Ph.D. dissertation [24], Ira Gessel proved a very general reciprocity formula which

allows one to obtain (as special cases) noncommutative symmetric function formulas counting

permutations with restrictions on the lengths of their increasing runs; we call this result the

“run theorem”. Our main result in this chapter is a generalization of Gessel’s run theorem

which allows for a much wider variety of restrictions on run lengths. Specifically, these

restrictions are those which can be encoded by a special type of digraph that we shall call a

“run network”.

We give two applications of the run theorem and generalized run theorem. Our first

application was inspired by a question posed by Liviu Nicolaescu on discrete Morse functions

arising in combinatorial topology, which is equivalent to the following: Let an be the number

of n-permutations with all peaks odd and all valleys even; what is the behavior of an/n! as

n→∞? We use the run theorem and generalized run theorem to find simple expressions for

the exponential generating functions counting permutations with parity restrictions on peaks

and valleys. The exponential generating function for the sequence {an}n≥0 turns out to be

an alternating analogue (in the sense of Sections 1.4 and 2.2) of a classical formula of David

and Barton, and we use this exponential generating function to derive an asymptotic formula

for an/n!, thus answering Nicolaescu’s question.

We also use the generalized run theorem to derive formulas for exponential generating

functions for polynomials of the form∑

π∈Sntst(π) for many permutation statistics st which

37

CHAPTER 3. THE RUN THEOREM AND ITS APPLICATIONS

are expressible in terms of increasing runs. These statistics include the peak number pk, right

peak number rpk, the double ascent number dasc, the number br of biruns, and the number

udr of up-down runs. Although equivalent formulas for some of these generating functions

have been discovered already using other methods, the generalized run theorem provides a

straightforward, systematic method for obtaining these generating function formulas.

3.2. Gessel’s run theorem

In this section and the following section, let A be a unital F -algebra of characteristic zero.

Theorem 3.1 (Run theorem). Let {w1, w2, . . . } be a set of weights from A, and for a

composition L = (L1, . . . , Lk), let wL = wL1wL2 · · ·wLk. Then, the noncommutative generating

function for words in P∗, in which each word with descent composition L is weighted wL, is∑L

wLrL =

( ∞∑n=0

vnhn

)−1

where the sum on the left is over all compositions L and the vn are defined by∞∑n=0

vnxn =

( ∞∑k=0

wkxk

)−1

(16)

with w0 = 1 (the unity element of A).

This theorem appeared in its original form as Theorem 5.2 of Gessel [24], and is similar

to Theorem 4.1 of Jackson and Aleliunas [31] and Theorem 4.2.3 of Goulden and Jackson

[26]. Gessel’s statement of the theorem does not explicitly use noncommutative symmetric

functions, which were not formally defined until 1995 in the seminal paper [18] of Gelfand

et al. However, Gessel and the present author [23, Theorem 11] restated the run theorem

using noncommutative symmetric functions and gave a different proof of the result, which we

present below.

38


Proof. Let us set vn = −un for n > 0, and for a composition K = (K1, . . . , Kk), let

uK = uK1 · · ·uKk. Then ( ∞∑

n=0

vnhn

)−1

=(

1−∞∑n=1

unhn

)−1

=∑K

uKhK

=∑K

uK∑L≤K

rL

=∑L

rL∑K≥L

uK . (17)

By (16), we have∞∑k=0

wkxn =

(1−

∞∑n=1

unxn)−1

so

wk =∑K�k

uK .

and thus

wL =∑K≥L

uK . (18)

Then the theorem follows from (17) and (18). �

Both previous versions of the run theorem—[24, Theorem 5.2] and [23, Theorem

11]—stated that the weights are to commute with each other, but the above proof does not

actually use this condition. Hence, we allow our algebra A to be commutative or noncommu-

tative.1 Although we can simply set A = Q in our applications, the fact that we can take

A to be noncommutative is pivotal to our proof of the generalized run theorem in the next

section.1We do, however, require that the weights commute with noncommutative symmetric functions. Formally,this means that we are working in the tensor product algebra A⊗F Sym.

39


Before continuing, we use the run theorem to prove Lemma 2.4, which says for any positive

integer m, we have the formula∑L

rL =( ∞∑n=0

(hmn − hmn+1))−1

where the sum on the left is over all compositions L with all parts less than m. This formula

will play an important role in our enumeration of permutations with all peaks odd and all

valleys even.

Proof of Lemma 2.4. We apply the run theorem with weights wi = 1 for i < m and

wi = 0 for i ≥ m. We have∞∑n=0

wnxn =

1− xm

1− x,

so∞∑n=0

vnxn =

1− x1− xm

=∞∑n=0

(xmn − xmn+1).

Then the result follows from the run theorem. �

3.3. The generalized run theorem

Suppose that G is a digraph on the vertex set [m], where each arc (i, j) is assigned

a nonempty subset Pi,j of P, and let P be the set of all pairs (a, e) where e = (i, j) is

an arc of G and a ∈ Pi,j. In addition, let−→P ∗ ⊆ P ∗ be the subset of all sequences α =

(a1, e1)(a2, e2) · · · (an, en) where e1e2 · · · en is a walk in G. Given α = (a1, e1)(a2, e2) · · · (an, en)

in−→P ∗, let ρ(α) = (a1, a2, . . . , an), and let E(α) = (i, j) where i and j are the initial and

terminal vertices, respectively, of the walk e1e2 · · · en.

We call this construction (G,P ) a run network if for all nonempty α, β ∈−→P ∗, if ρ(α) = ρ(β)

and E(α) = E(β) then α = β. That is, the same tuple (a1, a2, . . . , an) cannot be obtained

by traversing two different walks with the same initial and terminal vertices. Given a run

network (G,P ), suppose that we want the noncommutative generating function for words in

40


P∗ with descent composition L whose parts are given by a walk in G, with various weights

attached. This can be done using the following generalization of the run theorem.

Theorem 3.2 (Generalized run theorem). Suppose that G is a digraph on [m] and that

(G,P ) is a run network. Let {w(i.j)a : (a, (i, j)) ∈ P } be a set of weights from A, with w(i,j)

a = 0

if (a, (i, j)) /∈ P . Given a composition L and 1 ≤ i, j ≤ m, let w(i,j)(L) = we1L1· · ·wekLk

if

there exists α = (L1, e1) · · · (Lk, ek) ∈−→P ∗ such that E(α) = (i, j) and L = ρ(α), and let

w(i,j)(L) = 0 otherwise. Then,

∑L

w(1,1)(L)rL · · ·∑L

w(1,m)(L)rL

... . . . ...∑L

w(m,1)(L)rL · · ·∑L

w(m,m)(L)rL

=

∞∑n=0

v(1,1)n hn · · ·

∞∑n=0

v(1,m)n hn

... . . . ...∞∑n=0

v(m,1)n hn · · ·

∞∑n=0

v(m,m)n hn

−1

where each sum in the matrix on the left-hand side is over all compositions L and the v(i,j)n

are given by

∞∑n=0

v(1,1)n xn · · ·

∞∑n=0

v(1,m)n xn

... . . . ...∞∑n=0

v(m,1)n xn · · ·

∞∑n=0

v(m,m)n xn

=

Im +

∞∑k=1

w(1,1)k xk · · ·

∞∑k=1

w(1,m)k xk

... . . . ...∞∑k=1

w(m,1)k xk · · ·

∞∑k=1

w(m,m)k xk

−1

.

Proof. We apply the run theorem (Theorem 3.1) with weights coming from the algebra

Matm(A) of m×m matrices with entries in A. Set

wk =

w

(1,1)k · · · w

(1,m)k

... . . . ...

w(m,1)k · · · w

(m,m)k

and wL =

w(1,1)(L) · · · w(1,m)(L)

... . . . ...

w(m,1)(L) · · · w(m,m)(L)

.

41


Figure 1. Example of a run network

1 2

{2}

{3}

It suffices to verify that if L = (L1, . . . , Lk) is a composition, then wL = wL1wL2 · · ·wLk.

Indeed, the (i, j)th entry of wL1wL2 · · ·wLkis∑

1≤p1,...,pk+1≤mp1=i, pk+1=j

w(p1,p2)L1

w(p2,p3)L2

· · ·w(pk,pk+1)Lk

,

but at most one of these summands is nonzero because, by definition of a run network, no

composition can be obtained by traversing multiple walks with the same initial and terminal

vertices. This precisely gives us w(i,j)(L), the (i, j)th entry of wL, and we are done. �

For example, suppose that we want to count words having descent compositions of the form

(2, 3, 2, 3 . . . , 2, 3). Then, consider the run network in Figure 1; these descent compositions

correspond to walks in this digraph beginning and ending at vertex 1. By taking all nonzero

weights to be 1 and applying Theorem 3.2, it follows that the desired generating function is

the (1, 1) entry of the matrix ∞∑n=0

v(1,1)n hn

∞∑n=0

v(1,2)n hn

∞∑n=0

v(2,1)n hn

∞∑n=0

v(2,2)n hn

−1

where the v(i,j)n are given by

∞∑n=0

v(1,1)n xn

∞∑n=0

v(1,2)n xn

∞∑n=0

v(2,1)n xn

∞∑n=0

v(2,2)n xn

=

I2 +

0 x2

x3 0

−1

42


=

1

1− x5− x2

1− x5

− x3

1− x5

1

1− x5

=

∞∑n=0

x5 −∞∑n=0

x5n+2

−∞∑n=0

x5n+3

∞∑n=0

x5

. (19)

Therefore, our desired matrix is∞∑n=0

v(1,1)n hn

∞∑n=0

v(1,2)n hn

∞∑n=0

v(2,1)n hn

∞∑n=0

v(2,2)n hn

−1

=

∞∑n=0

h5n −∞∑n=0

h5n+2

−∞∑n=0

h5n+3

∞∑n=0

h5n

−1

. (20)

Now, suppose that we wish to count permutations with descent compositions of the form

(2, 3, 2, 3 . . . , 2, 3). The homomorphism Φ defined in Section 2.2 induces a homomorphism on

the corresponding matrix algebras, which we also call Φ by a slight abuse of notation. Then,

by applying Φ to (20), we obtain the matrix∞∑n=0

x5n

(5n)!−∞∑n=0

x5n+2

(5n+ 2)!

−∞∑n=0

x5n+3

(5n+ 3)!

∞∑n=0

x5n

(5n)!

−1

, (21)

whose (1, 1) entry is the exponential generating function for permutations having descent

compositions of the form (2, 3, 2, 3 . . . , 2, 3). Observe that (21) is the inverse of the matrix

obtained by taking (19) and converting the ordinary generating functions to exponential

generating functions.

If we apply the homomorphism Φ to (20) instead of Φ, then we would obtain the analogous

result for alternating runs. By applying Φq instead, we would obtain a q-exponential generating

function which also keeps track of the inversion number inv.

43


Finally, we note that the original run theorem can be recovered from the generalized run

theorem by using the run network with one vertex and a loop to which the entire set P is

assigned. Hence, Theorem 3.2 is indeed a generalization of Theorem 3.1.

3.4. Permutations with parity restrictions on peaks and valleys

3.4.1. Statement of main result. In this section, we enumerate permutations with

parity restrictions on peaks and valleys. A priori, there are four cases to consider, but it

is clear that permutations with all peaks odd and all valleys even are in bijection with

permutations with all peaks even and all valleys odd via complementation, so there are

actually only three cases.

Recall that En is the nth Euler number defined by∑∞

n=0 Enxn/n! = sec x+ tanx.

Theorem 3.3. Let an be the number of n-permutations with all peaks odd and all valleys

even, let bn be the number of n-permutations with all peaks and valleys even, and let cn be the

number of n-permutations with all peaks and valleys odd.

(a) The exponential generating function A(x) for {an}n≥0 is

A(x) =

(1− E1x+ E3

x3

3!− E4

x4

4!+ E6

x6

6!− E7

x7

7!+ · · ·

)−1

=3 sin

(12x)

+ 3 cosh(√

32x)

3 cos(

12x)−√

3 sinh(√

32x) .

(b) The exponential generating function B(x) for {bn}n≥0 is

B(x) = (1 + x)2 + 2 cosh(

√2x) +

√2x sinh(

√2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x)

(c) The exponential generating function C(x) for {cn}n≥0 is

C(x) =2 + 2 cosh(

√2x) +

√2(2 + x) sinh(

√2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x).

44


Table 1. The sequences {an}n≥0, {bn}n≥0, and {cn}n≥0

n 0 1 2 3 4 5 6 7 8 9 10 11 12

an 1 1 2 4 13 50 229 1238 7614 52706 405581 3432022 31684445

bn 1 1 2 6 8 40 84 588 1632 14688 51040 561440 2340480

cn 1 1 2 2 8 14 84 204 1632 5104 51040 195040 2340480

The first several terms of these sequences are given in Table 1. The sequence {an}n≥0

can be found on the OEIS [48, A246012]. Observe that b2n = c2n for all n; this is immediate

from the fact that 2n-permutations with all peaks and valleys odd are in bijection with

2n-permutations with all peaks and valleys even via reversion.

3.4.2. All peaks odd and all valleys even. Here we prove part (a) of Theorem 3.3.

Our proof involves the notion of alternating runs (defined in Section 1.4) and uses the

homomorphism Φ (defined in Section 2.2).

Lemma 3.4. A permutation has all peaks odd and all valleys even if and only if all of its

alternating runs have length less than 3.

Proof. An even peak of a permutation π corresponds to a subsequence π2i−1 < π2i >

π2i+1 and thus will be contained in an alternating run of length at least 3, and similarly an

odd valley must also be contained in an alternating run of length at least 3. Conversely, any

alternating run of length at least 3 contains (as its second letter) either an even peak or an

odd valley. �

Now we are ready for the main proof.

Proof of Theorem 3.3 (a). By (8), we know that[∞∑n=0

(E3n

x3n

(3n)!− E3n+1

x3n+1

(3n+ 1)!

)]−1

45


is the exponential generating function for permutations with all alternating runs having

length less than 3, and in light of Lemma 3.4 these are precisely the permutations that we

are counting. Thus

A(x) =

[∞∑n=0

(E3n

x3n

(3n)!− E3n+1

x3n+1

(3n+ 1)!

)]−1

=

(1− E1x+ E3

x3

3!− E4

x4

4!+ E6

x6

6!− E7

x7

7!+ · · ·

)−1

.

Now, let E(x) := secx+ tanx. By multisection, we have∞∑n=0

E3nx3n

(3n)!=

1

3(E(x) + E(ωx) + E(ω2x))

and∞∑n=0

E3n+1x3n+1

(3n+ 1)!=

1

3(E(x) + ω−1E(ωx) + ω−2E(ω2x)),

where ω is the primitive cube root of unity e2πi/3. It can then be verified, for example by

using Maple, that

A(x) =

[∞∑n=0

(E3n

x3n

(3n)!− E3n+1

x3n+1

(3n+ 1)!

)]−1

(22)

=

[1

3(E(x) + E(ωx) + E(ω2x))− 1

3(E(x) + ω−1E(ωx) + ω−2E(ω2x))

]−1

=3 sin

(12x)

+ 3 cosh(√

32x)

3 cos(

12x)−√

3 sinh(√

32x) , (23)

which completes the proof. �

We note that the formula (23) for A(x) can be proven directly by solving a system of

differential equations; see [23, Section 2.1]. Although (22) is a more elegant formula, (23) is

better-suited for extracting the asymptotic data of A(x). The following is our asymptotic

formula for an/n!, which answers the question of Nicolaescu that originally inspired the work

in this section. The proof is omitted here but can be found in [23, Section 2.2].

46


Theorem 3.5. Let an be the number of n-permutations with all peaks odd and all valleys

even. Thenann!

= 2βn+1 +O(δn), as n→∞,

where β = 0.7693323708 · · · and δ = 0.3049639861 · · · .

3.4.3. All peaks and valleys even. We now find the exponential generating function

B(x) for permutations with all peaks and valleys even using the generalized run theorem.

Proof of Theorem 3.3 (b). In order to use the generalized run theorem, we need a

run network which encodes the possible descent compositions of permutations with all peaks

and valleys even. This can be done, but will depend on whether the permutation begins with

an ascent or descent and ends with an ascent or descent.

Notice that the permutations counted by B(x) which start and end with ascents are in

bijection via complementation with those permutations which start and end with descents. For

the same reason, those starting with a descent and ending with an ascent are equinumerous

with those starting with an ascent and ending with a descent. Thus, we only need to consider

two cases.

First, let us consider permutations with all peaks and valleys even which begin and end

with ascents. The descent compositions of these permutations, other than the increasing

permutations 12 · · ·n, are given by walks from vertex 1 to vertex 5 in the run network in

Figure 2, which we call (G1, P1). Indeed, the permutation must begin with an increasing

run of even length before reaching a peak, followed by an odd number of short runs before

reaching a valley.2 Then, going from a valley to a peak corresponds to a long run of odd

length, and once again followed by an odd number of short runs before reaching another

valley. This pattern continues until the permutation reaches its final valley, and then ends

with a long run.2Recall that an increasing run is called short if it has length 1, and it is called long if it has length at least 2.

47


Figure 2. Run network (G1, P1) for permutations with all peaks and valleyseven which begin and end with ascents

1 2 3 4

5

{2, 4, 6, . . . }

{1} {1}

{3, 5, 7, . . . }{2, 3, 4, 5, . . . }

{1}

Let B1(x) denote the exponential generating function for the permutations whose descent

compositions correspond to walks from 1 to 5 in the run network (G1, P1). Applying Theorem

3.2 with all nonzero weights set equal to 1 and then applying the homomorphism Φ, it follows

that B1(x) is the (1, 5) entry of the matrix

∞∑n=0

v(1,1)n

xn

n!· · ·

∞∑n=0

v(1,5)n

xn

n!... . . . ...

∞∑n=0

v(5,1)n

xn

n!· · ·

∞∑n=0

v(5,5)n

xn

n!

−1

,

where the v(i,j)n are given by

∞∑n=0

v(1,1)n xn · · ·

∞∑n=0

v(1,5)n xn

... . . . ...∞∑n=0

v(5,1)n xn · · ·

∞∑n=0

v(5,5)n xn

=

I5 +

0 x2

1−x2 0 0 0

0 0 x 0 0

0 x3

1−x2 0 x x2

1−x

0 0 x 0 0

0 0 0 0 0

−1

48


=

1 − (1−x2)x2

1−2x2x3

1−2x2− x4

1−2x2− x5

(1−2x2)(1−x)

0 1−2x2+x4

1−2x2(1−x2)x1−2x2

(1−x2)x2

1−2x2(1+x)x3

1−2x2

0 − x3

1−2x21−x21−2x2

(1−x2)x1−2x2

−x2(1+x)1−2x2

0 x4

1−2x2− (1−x2)x

1−2x2−1−2x2+x4

1−2x2x3(1+x)1−2x2

0 0 0 0 1

.

Converting these ordinary generating functions to exponential generating functions yields

1 1−x2−cosh(√

2x)4

−2x−√

2 sinh(√

2x)4

1+x2−cosh(√

2x)4

3+2x+x2+√

2 sinh(√

2x)+cosh(√

2x)−ex4

0 3−x2+cosh(√

2x)4

−2x+√

2 sinh(√

2x)4

−1−x2−cosh(√

2x)4

−1+2x+x2−√

2 sinh(√

2x)−cosh(√

2x)4

0 2x−√

2 sinh(√

2x)4

1+cosh(√

2x)2

−2x+√

2 sinh(√

2x)4

2+2x−√

2 sinh(√

2x)+2 cosh(√

2x)4

0 −1+x2−cosh(√

2x)4

−2x+√

2 sinh(√

2x)4

3+x2+cosh(√

2x)4

−1+2x+x2−√

2 sinh(√

2x)−cosh(√

2x)4

0 0 0 0 1

,

whose inverse matrix has (1, 5) entry

B1(x) =4 + 4x+ x2 + (4− x2 − 4ex) cosh(

√2x) + 2

√2(1 + xex) sinh(

√2x)− 4ex

4 + 4 cosh(√

2x)− 2√

2x sinh(√

2x).

Next, we consider permutations with all peaks and valleys even which begin with a descent

and end with an ascent. The increasing runs of these permutations follow a very similar

pattern as before, but it must begin with an odd number of short runs because the first letter

is a descent rather than an ascent. Therefore, their descent compositions are given by walks

from 1 to 5 in the run network in Figure 3, which we call (G2, P2).

Repeating the same procedure as before, we start by computingI5 +

0 x 0 0 0

0 0 x x3

1−x2x2

1−x

0 x 0 0 0

0 x 0 0 0

0 0 0 0 0

−1

=

1 − (1−x2)x1−2x2

(1−x2)x2

1−2x2x4

1−2x2(1+x)x3

1−2x2

0 1−x21−2x2

(1−x2)x1−2x2

− x3

1−2x2− (1+x)x2

1−2x2

0 − (1−x2)x1−2x2

1−x2−x41−2x2

x4

1−2x2(1+x)x3

1−2x2

0 − (1−x2)x1−2x2

(1−x2)x2

1−2x2(1−x2)2

1−2x2(1+x)x3

1−2x2

0 0 0 0 1

,

49


Figure 3. Run network (G2, P2) for permutations with all peaks and valleyseven which begin with a descent and end with an ascent

1 2

3

4

5{1}

{1}

{3, 5, 7, . . . }

{2, 3, 4, 5, . . . }

{1}

{1}

and then we convert the ordinary generating functions to exponential generating functions to

obtain

1 −2x+√

2 sinh(√

2x)4


2x)4

−1+x2−cosh(√

2x)4

−1+2x+x2−√

2 sinh(√

2x)−cosh(√

2x)4

0 1+cosh(√

2x)2

−2x+√

2 sinh(√

2x)4

2x−√

2 sinh(√

2x)4

2+2x−√

2 sinh(√

2x)−2 cosh(√

2x)4

0 −2x+√

2 sinh(√

2x)4

3+x2+cosh(√

2x)4

−1+x2−cosh(√

2x)4

−1+2x+x2−√

2 sinh(√

2x)−cosh(√

2x)4

0 −2x+√

2 sinh(√

2x)4


2x)4

3−x2+cosh(√

2x)4

−1+2x+x2−√

2 sinh(√

2x)−cosh(√

2x)4

0 0 0 0 1

.

The (1, 5) entry of the inverse matrix gives us

B2(x) =x2 − x(4 + x) cosh(

√2x) + 2

√2 sinh(

√2x)

4 + 4 cosh(√

2x)− 2√

2x sinh(√

2x).

Now we can obtain B(x) by taking 2B1(x) + 2B2(x), but will need to add an additional

term to account for the increasing and decreasing permutations which were excluded from

the above computations. Since ex is the exponential generating function for increasing

50


permutations and also for decreasing permutations, we add 2ex but also substract x + 1

because the empty permutation and the length 1 permutation are counted twice by 2ex.

Therefore,

B(x) = 2B1(x) + 2B2(x) + 2ex − x− 1

which simplifies to

B(x) = (1 + x)2 + 2 cosh(

√2x) +

√2x sinh(

√2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x)

as desired. �

3.4.4. All peaks and valleys odd. Although the exponential generating function C(x)

for permutations with all peaks and valleys odd can be obtained in the same way as B(x) via

the generalized run theorem, it is easier to derive it from a combinatorial identity relating

the odd and even terms of the sequence {cn}n≥0, which we do here.

Notice that the generating function B(x) for permutations with all peaks and valleys even

splits nicely into even and odd parts:

Beven(x) :=∞∑n=0

b2nx2n

(2n)!=

2 + 2 cosh(√

2x) +√

2x sinh(√

2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x)

and

Bodd(x) :=∞∑n=0

b2n+1x2n+1

(2n+ 1)!= x

2 + 2 cosh(√

2x) +√

2x sinh(√

2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x).

We immediately deduce from Bodd(x) = xBeven(x) an identity relating the even and odd

terms of {bn}n≥0.

Lemma 3.6. For all n ≥ 0, b2n+1 = (2n+ 1)b2n.

This identity can also be seen from a simple bijection.

Proof. Let π be any 2n-permutation with all peaks and valleys even, and pick any

m ∈ [2n + 1]. Let π′ be the permutation obtained by replacing the letter k with k + 1 for

51


every k ≥ m in π, and appending m to the end of π. For example, given π = 1432 and m = 3,

we have π′ = 15423. Then, π′ is a (2n+ 1)-permutation with all peaks and valleys even. To

obtain (π,m) from π′, simply take m = π′2n+1 and standardize the word formed by the first

2n letters of π′ to get π ∈ S2n. �

Essentially the same bijection gives us an analogous identity for permutations with all

peaks and valleys odd.

Lemma 3.7. For all n ≥ 1, c2n = 2nc2n−1.

Proof. Pick any (2n− 1)-permutation with all peaks and valleys odd, and any m ∈ [2n].

Applying the same procedure in the proof of Lemma 3.6 yields a 2n-permutation with all

peaks and valleys odd, and we reverse the procedure in the same way to obtain π and m. �

Now we complete the proof of Theorem 3.3 (c).

Proof of Theorem 3.3 (c). Lemma 3.7, along with the fact that b2n = c2n for all

n ≥ 0, allows us to deduce that

Codd(x) :=∞∑n=1

c2n−1x2n−1

(2n− 1)!

=∞∑n=1

c2n

2n

x2n−1

(2n− 1)!

=1

x

∞∑n=1

b2nx2n

(2n)!

=1

x

(2 + 2 cosh(

√2x) +

√2x sinh(

√2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x)− 1

)

=2√

2 sinh(√

2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x).

Furthermore,

Ceven(x) :=∞∑n=0

c2nx2n

(2n)!= Beven(x),

52


so we have

C(x) = Codd(x) + Ceven(x)

=2 + 2 cosh(

√2x) +

√2(2 + x) sinh(

√2x)

2 + 2 cosh(√

2x)−√

2x sinh(√

2x). �

Permutations with all peaks and valleys odd are closely related to “balanced permutations”,

which are defined in terms of standard skew Young tableaux called “balanced tableaux”. In

fact, balanced permutations of odd length are precisely permutations of odd length with all

peaks and valleys odd, counted by {c2n+1}n≥0. Gessel and Greene [21] gave the exponential

generating function for balanced permutations, and by comparing generating functions,

showed that

d2n+1 = 2nc2n+1 (24)

for all n ≥ 0, where dn is the number of n-permutations with all valleys odd (and with no

parity restrictions on peaks) which were previously studied by Gessel [20]. A bijective proof

of (24) was later given by La Croix [35].

3.5. Counting permutations by run-expressible descent statistics

3.5.1. General setup. Given a permutation statistic st, we define the polynomial P stn (t)

by

P stn (t) :=

∑π∈Sn

tst(π).

The exponential generating function

P st(t, x) :=∞∑n=0

P stn (t)

xn

n!

is a bivariate generating function counting permutations by the statistic st.

In this section, we compute the generating functions P st(t, x) for various descent statistics

st by applying the generalized run theorem on the run network in Figure 4, which we call

53


Figure 4. Run network (G,P ) for counting permutations by run-expressiblestatistics

1 2 3P

P

P

(G,P ). The descent compositions of all non-increasing permutations are given by walks from

1 to 3 in (G,P ), but note that we distinguish the initial run and the final run in this run

network. The descent statistics that we consider in this section are all determined by the

number of non-initial and non-final long runs, as well as whether the permutation begins

and ends with a short run or a long run. Hence, by assigning appropriate weights to the

letters of P in this run network, the generalized run theorem yields refined results counting

permutations by these statistics.

Not every statistic that we consider requires three vertices in our run network. For

example, only two vertices are required if we only need to distinguish the initial run or the

final run but not both, and only one vertex is required if we do not need to distinguish

the initial run or the final run. We will still use the 3-vertex run network (G,P ) in the

former case, since it eliminates the need for us to define two 2-vertex run networks (one

for distinguished initial runs, and one for distinguished final runs) and the computation is

no more difficult using a computer algebra system such as Maple. However, the latter case

does not require a run network at all, so we will simply apply the original version of the run

theorem (Theorem 3.1).

This general approach can also be used to find multivariate generating functions giving

the joint distribution of two or more of these statistics, although we do not do this here.

We note that the result of applying the generalized run theorem to the 3-vertex run

network (G,P ) is essentially a weighted version of Theorem 6.12 of Gessel [24], which can be

54


Table 2. pk polynomials

n P pkn (t) n P pk

n (t)

0 1 5 16 + 88t+ 16t2

1 1 6 32 + 416t+ 272t2

2 2 7 64 + 1824t+ 2880t2 + 272t3

3 4 + 2t 8 128 + 7680t+ 24576t2 + 7936t3

4 8 + 16t 9 256 + 31616t+ 185856t2 + 137216t3 + 7936t4

used to obtain formulas for counting words with distinguished initial run and final run, and

is similar to results given by Jackson and Aleliunas [31, Sections 10–12]. See also Goulden

and Jackson [26, Theorem 4.2.19].

3.5.2. Counting by peaks (and variations). The first two statistics that we consider

are the peak number pk and valley number val of a permutation. By taking complements,

we see that these two statistics are equidistributed over Sn, so it suffices to find the bivariate

generating function for pk.

Theorem 3.8.

P pk(t, x) =

√1− t cosh(x

√1− t)√

1− t cosh(x√

1− t)− sinh(x√

1− t)

Other equivalent formulas have been found, e.g., by Entringer [14] using differential

equations, Mendes and Remmel [39] using the “homomorphism method”, and Kitaev [33]

using the notion of partially ordered permutation patterns.

The first ten polynomials P pkn (t) are given in Table 2, and their coefficients can be found

in the OEIS [48, A008303].

Proof. By Lemma 1.3 (a), the number of peaks in a permutation is equal to its number

of non-final long runs. So, we want to weight every k 6= 1 in P1,2 and P2,2 by t in (G,P ).

55


Setting w(1,2)k = w

(2,2)k = t for all k 6= 1 (and setting all other nonzero weights to 1) and then

applying Theorem 3.2, we computeI3 +

0 x+ tx2

1−x 0

0 x+ tx2

1−x x+ x2

1−x

0 0 0

−1

=

1 − (1−(1−t)x)x

1−(1−t)x2(1−(1−t)x)x2

(1−(1−t)x2)(1−x)

0 1−x1−(1−t)x2 − x

1−(1−t)x2

0 0 1

,

and converting the ordinary generating functions to exponential generating functions gives1 −1 + cosh(x

√1− t)− sinh(x

√1−t)√

1−t −1− sinh(x√

1−t)√1−t + ex

0 cosh(x√


1−t)√1−t − sinh(x

√1−t)√

1−t

0 0 1

. (25)

Since the increasing permutations have exponential generating function ex and do not have

any peaks, we add ex to the (1, 3) entry of the inverse matrix of (25) to obtain our desired

generating function

P pk(t, x) =

√1− t cosh(x

√1− t)√

1− t cosh(x√


1− t). �

Now, let us consider the analogous result for the left peak number lpk and right peak

number rpk. Since lpk and rpk are equidistributed over Sn by reversion, we only need to find

the bivariate generating function for one of these statistics, say, rpk. (We can also consider

the number of “left valleys” and the number of “right valleys” defined in the obvious way, but

these are also equidistributed over Sn along with lpk and rpk.)

Theorem 3.9.

P rpk(t, x) =

√1− t√

1− t cosh(x√


1− t)

The first ten polynomials P rpkn (t) are given in Table 3, and their coefficients can also be

found in the OEIS [48, A008971].

56


Table 3. rpk polynomials

n P rpkn (t) n P rpk

n (t)

0 1 5 1 + 58t+ 61t2

1 1 6 1 + 179t+ 479t2 + 61t3

2 1 + t 7 1 + 543t+ 3111t2 + 1385t3

3 1 + 5t 8 1 + 1636t+ 18270t2 + 19028t3 + 1385t4

4 1 + 18t+ 5t2 9 1 + 4916t+ 101166t2 + 206276t3 + 50521t4

Proof. The number of right peaks of a permutation is equal to its total number of long

runs by Lemma 1.3 (d), so we now assign a weight t to every such run. Thus, setting wk = t

for all k 6= 1 and applying the original run theorem, we have that(1 + x+

tx2

1− x

)−1

=1− x

1− (1− t)x2,

whose coefficients have exponential generating function

cosh(x√


1− t)√1− t

.

Finally, taking the reciprocal yields

P rpk(t, x) =

(cosh(x

√1− t)− sinh(x

√1− t)√

1− t

)−1

=

√1− t√

1− t cosh(x√


1− t). �

By comparing P pk(t, x) and P rpk(t, x), we obtain the following formulas.

Corollary 3.10.

P pk(t, x) = P rpk(t, x) cosh(x√

1− t),

or equivalently,

P rpk(t, x) = P pk(t, x) sech(x√

1− t).

57


We do not know of a combinatorial explanation for this fact. Note that some of the

coefficients of

cosh(x√

1− t) =∞∑n=0

(1− t)n x2n

(2n)!

are negative, so there may be some sort of inclusion-exclusion phenomenon at play. A

combinatorial explanation may also involve alternating permutations because of the formula

sech(x√

1− t) =∞∑n=0

E2n(t− 1)nx2n

(2n)!.

Finally, below is the result for the exterior peak number. The polynomials P epkn (t) are

essentially the polynomials P valn (t) since epk(π) = val(π) + 1; see Lemma 1.3 (e).

Theorem 3.11.

P epk(t, x) =

√1− t cosh(x

√1− t)− (1− t) sinh(x

√1− t)√

1− t cosh(x√


1− t)

Proof. Since epk(π) = val(π) + 1 and pk is equidistributed with val over Sn, it follows

from Theorem 3.8 that

P epk(t, x) = tP pk(t, x)− t+ 1

=

√1− t cosh(x

√1− t)− (1− t) sinh(x

√1− t)√

1− t cosh(x√


1− t). �

3.5.3. Counting by double ascents (and variations). Our next result gives a bi-

variate generating function counting permutations by the double ascent number dasc.

Theorem 3.12.

P dasc(t, x) =ue

12

(1−t)x

u cosh(12ux)− (1 + t) sinh(1

2ux)

where u =√

(t+ 3)(t− 1).

58


Table 4. dasc polynomials

n P dascn (t) n P dasc

n (t)

0 1 5 70 + 41t+ 8t2 + t3

1 1 6 349 + 274t+ 86t2 + 10t3 + t4

2 2 7 2017 + 2040t+ 803t2 + 167t3 + 12t4 + t5

3 5 + t 8 13358 + 16346t+ 8221t2 + 2064t3 + 316t4 + 14t5 + t6

4 17 + 6t+ t2 9 99377 + 143571t+ 86214t2 + 28143t3 + 4961t4 + 597t5 + 16t6 + t7

The first ten polynomials P dascn (t) are given in Table 4; see also their OEIS entry [48,

A162975].

Proof. It is clear that short runs contribute no double ascents, and long runs of length

k ≥ 2 contribute k − 2 double ascents. Thus, we set wk = tk−2 for all k 6= 1 and apply the

original run theorem to obtain(1 + x+

x2

1− tx

)−1

=1− tx

1 + (1− t)(1 + x)x,

whose coefficients have exponential generating function

e−12

(1−t)x(

cosh(1

2ux)− (1 + t)

usinh

(1

2ux

))where u =

√(t+ 3)(t− 1). Then taking the reciprocal gives us

P dasc(t, x) =

(e−

12

(1−t)x(

cosh(1

2ux)− (1 + t)

usinh

(1

2ux)))−1

=ue

12

(1−t)x


2ux)

. �

We note that Elizalde and Noy, in their study of consecutive permutation patterns [13],

previously found the formula

P dasc(t, x) =2ue

12

(1−t+u)x

1 + t+ u+ eux(u− t− 1)

59


Table 5. rdasc polynomials

n P rdascn (t)

0 1

1 1

2 1 + t

3 3 + 2t+ t2

4 9 + 11t+ 3t2 + t3

5 39 + 48t+ 28t2 + 4t3 + t4

6 189 + 297t+ 166t2 + 62t3 + 5t4 + t5

7 1107 + 1902t+ 1419t2 + 476t3 + 129t4 + 6t5 + t6

8 7281 + 14391t+ 11637t2 + 5507t3 + 1235t4 + 261t5 + 7t6 + t7

9 54351 + 118044t+ 111438t2 + 56400t3 + 19096t4 + 3020t5 + 522t6 + 8t7 + t8

where again u =√

(t+ 3)(t− 1). Their formula for P dasc(t, x) is equivalent to our formula

from Theorem 3.12.

Now, let us consider the right double ascent number rdasc and the exterior double ascent

number edasc.

Theorem 3.13.

P rdasc(t, x) =u cosh(1

2ux) + (1− t) sinh(1

2ux)


2ux)

and

P edasc(t, x) =ue−

12

(1−t)x


2ux)

where u =√

(t+ 3)(t− 1).

The first ten of the polynomials P rdascn (t) and P edasc

n (t) are given in Tables 5 and 6,

respectively. There is an OEIS entry [48, A162976] for the coefficients of the P rdascn (t), but

there does not seem to be one for the P edascn (t).

60


Table 6. edasc polynomials

n P edascn (t)

0 1

1 t

2 1 + t2

3 1 + 4t+ t3

4 6 + 6t+ 11t2 + t4

5 19 + 51t+ 23t2 + 26t3 + t5

6 109 + 212t+ 269t2 + 72t3 + 57t4 + t6

7 588 + 1571t+ 1419t2 + 1140t3 + 201t4 + 120t5 + t7

8 4033 + 10470t+ 13343t2 + 7432t3 + 4272t4 + 522t5 + 247t6 + t8

9 29485 + 87672t+ 107853t2 + 87552t3 + 33683t4 + 14841t5 + 1291t6 + 502t7 + t9

Proof. As before, non-final short runs contribute no right double ascents, and non-final

long runs of length k ≥ 2 contribute k − 2 right double ascents. Moreover, if the final

increasing run is of length k, then it contributes k − 1 right double ascents. So, we take

w(1,2)k = w

(2,2)k = tk−2 for all k 6= 1 and w(2,3)

k = tk−1 for all k in the same run network (G,P )

defined earlier, and applying Theorem 3.2 givesI3 +

0 x+ x2

1−tx 0

0 x+ x2

1−txx

1−tx

0 0 0

−1

=

1 − (1+(1−t)x)x

1+x(1−t)(1+x)(1+(1−t)x)x2

(1+(1−t)x2)(1−tx)

0 1−tx1+x(1−t)(1+x)

− x1+x(1−t)(1+x)

0 0 1

,

and converting to exponential generating functions gives1 −1 + e−

12

(1−t)x(cosh(12ux)− 1+t

usinh(1

2ux))− 2ue−

12

(1−t) sinh(12ux)− 1−etx

t

0 e−12

(1−t)x(cosh(12ux)− 1+t

usinh(1

2ux))

− 2ue−

12

(1−t) sinh(12ux)

0 0 1

.where u =

√(t+ 3)(t− 1).

61


We still need to account for the increasing permutations, and the increasing permutation

of length n has n− 1 right double ascents. So, we take the (1, 3) entry of the inverse of the

above matrix and add to it (etx − 1)/t+ 1 to obtain our desired generating function

P rdasc(t, x) =u cosh(1

2ux) + (1− t) sinh(1

2ux)


2ux)

.

The computation for exterior double ascents is similar, but we have to adjust the weights

for the initial increasing run. If the initial increasing run is of length k, then it contributes k−1

left-right double ascents; hence, we take w(2,2)k = tk−2 for all k 6= 1 and w(1,2)

k = w(2,3)k = tk−1

for all k. Then the computation proceeds in the same way, and we add etx at the end because

the increasing permutation of length n has n exterior double ascents. �

Comparing our expressions for P dasc(t, x) and P edasc(t, x) gives the following formula.

Corollary 3.14.

P dasc(t, x) = e(1−t)xP edasc(t, x)

We do not know of a combinatorial proof.

3.5.4. Counting by biruns and up-down runs. Finally, we compute bivariate gen-

erating functions counting permutations by the number of biruns br and the number of

up-down runs udr.

Theorem 3.15.

P br(t, x) =v

(1 + t)2· 2t+ (1 + x+ t2(1− x)) cosh(vx)− v(1 + x) sinh(vx)

v cosh(vx)− sinh(vx)

and

P udr(t, x) =v

1 + t· t+ cosh(vx)− v sinh(vx)

v cosh(vx)− sinh(vx)

where v =√

1− t2.

62


Table 7. br polynomials

n P brn (t)

0 1

1 1

2 2t

3 2t+ 4t2

4 2t+ 12t2 + 10t3

5 2t+ 28t2 + 58t3 + 32t4

6 2t+ 60t2 + 236t3 + 300t4 + 122t5

7 2t+ 124t2 + 836t3 + 1852t4 + 1682t5 + 544t6

8 2t+ 252t2 + 2766t3 + 9576t4 + 14622t5 + 10332t6 + 2770t7

9 2t+ 508t2 + 8814t3 + 45096t4 + 103326t5 + 119964t6 + 69298t7 + 15872t8

Table 8. udr polynomials

n P udrn (t)

0 1

1 t

2 t+ t2

3 t+ 3t2 + 2t3

4 t+ 7t2 + 11t3 + 5t4

5 t+ 15t2 + 43t3 + 45t4 + 16t5

6 t+ 31t2 + 148t3 + 268t4 + 211t5 + 61t6

7 t+ 63t2 + 480t3 + 1344t4 + 1767t5 + 1113t6 + 272t7

8 t+ 127t2 + 1509t3 + 6171t4 + 12099t5 + 12477t6 + 6551t7 + 1385t8

9 t+ 255t2 + 4661t3 + 26955t4 + 74211t5 + 111645t6 + 94631t7 + 42585t8 + 7936t9

The first ten of the polynomials P brn (t) and P udr

n (t) are given in Tables 7 and 8, respectively.

Also see their OEIS entries [48, A059427 and A186370].

63


Proof. Recall from Lemma 1.3 (e) that br(π) = pk(π)+val(π)+1. Moreover, by Lemma

1.3 (a) and (b), the number of peaks in a permutation is equal to its number of non-final long

runs and that the number of valleys is equal to its number of non-initial long runs. Hence,

using the run network (G,P ) as before, we set w(1,2)k = w

(2,3)k = t and w(2,2)

k = t2 for all k 6= 1.

Then, I3 +

0 x+ tx2

1−x 0

0 x+ t2x2

1−x x+ tx2

1−x

0 0 0

−1

=

1 − (1−(1−t)x)x

1−(1−t2)x2(1−(1−t)x)2x2

(1−(1−t2)x2)(1−x)

0 1−x1−(1−t2)x2

− (1−(1−t)x)x1−(1−t2)x2

0 0 1

,and converting to exponential generating functions gives

1 −1−cosh(vx)1+t

− sinh(vx)v

−1 +(

1−t1+t

)x− 2 sinh(vx)

(1+t)v+ ex

0 cosh(vx)− sinh(vx)v

−1−cosh(vx)1+t

− sinh(vx)v

0 0 1

where v =

√1− t2. Finally, we take the (1, 3) entry of the inverse matrix, add ex to account

for the increasing permutations, multiply by t, and then add −tx− t+ x+ 1. The result is

P br(t, x) =v

(1 + t)2· 2t+ (1 + x+ t2(1− x)) cosh(vx)− v(1 + x) sinh(vx)

u cosh(vx)− sinh(vx),

as stated.

To compute the bivariate generating function P udr(t, x) for the number of up-down runs,

we use the same weights as before but also weight initial short runs. That is, we set w(1,2)k = t

for all k, and set w(2,2)k = t2 and w(2,3)

k = t for all k 6= 1. Then the computation is done in the

same way, and at the end we add ex, multiply by t, and add −t + 1 to obtain the desired

generating function. �

A formula equivalent to our above formula for P udr(t, x) was found earlier by Richard Stanley

in his study of longest alternating subsequences of permutations. Let as(π) be the length of the

longest alternating subsequence of a permutation π. Then, for example, the n-permutations

64


π with as(π) = n are the length n alternating permutations. The number of up-down runs in

a permutation is equal to the length of its longest alternating subsequence; an alternating

subsequence is obtained by taking the last letter of each up-down run, and it is easy to see

that this is indeed a longest alternating subsequence. For example, the up-down runs of

π = 51378624 are 5, 51, 1378, 862, and 24, so 51824 is a longest alternating subsequence

of π, which has length equal to the number of up-down runs of π. Stanley [53] derived the

bivariate generating function

P as(t, x) = (1− t) 1 + v + 2tevx + (1− v)e2vx

1 + v − t2 + (1− v − t2)e2vx

where v =√

1− t2, and gave the identity

P asn (t) =

(1 + t

2

)P brn (t) (26)

for n ≥ 2. Other properties of the udr statistic can be deduced by studying the as statistic,

and vice versa.

65

CHAPTER 4

Eulerian polynomials and descent statistics

4.1. Introduction

Recall that the nth Eulerian polynomial is defined by

An(t) :=∑π∈Sn

tdes(π)+1

for n ≥ 1 and by A0(t) := 1; these polynomials encode the distribution of the descent number

over Sn. The Eulerian polynomials also appear in a number of formulas for polynomials that

encode the distributions of other descent statistics. Here are several known results from the

literature:

• The pk polynomials1 P pkn (t) defined by

P pkn (t) :=

∑π∈Sn

tpk(π)+1

for n ≥ 1 and by P pk0 (t) := 1 are related to the Eulerian polynomials by the identity

An(t) =

(1 + t

2

)n+1

P pkn

(4t

(1 + t)2

)(27)

for n ≥ 1, which was first stated explicitly by Stembridge [58] as a result of his

theory of enriched P -partitions. However, (27) can also be proven using an earlier

construction of Shapiro, Woan, and Getu [47] which was later rediscovered by

Brändén as the “modified Foata–Strehl action” [4], a variant of a group action on

permutations originally defined by Foata and Strehl [17]. Making a substitution1Note that the definition of P pk

n (t) given here is slightly different than the definition given in Chapter 3.

66

CHAPTER 4. EULERIAN POLYNOMIALS AND DESCENT STATISTICS

yields the equivalent identity

P pkn (t) =

(2

1 + v

)n+1

An(v)

where v = 2t(1−

√1− t)− 1.

• The lpk polynomials

P lpkn (t) :=

∑π∈Sn

tlpk(π)

are related to the Eulerian polynomials by the identityn∑k=0

(n

k

)2k(1− t)n−kAk(t) = (1 + t)nP lpk

n

(4t

(1 + t)2

)(28)

for all n. This identity was proven by Petersen [41, Observation 3.1.2] using a

modification of enriched P -partitions called “left enriched P -partitions”. Equivalently,

P lpkn (t) =

1

(1 + v)n

n∑k=0

(n

k

)2k(1− v)n−kAk(v)

where again v = 2t(1−

√1− t)− 1.

• The br polynomials

P brn (t) :=

∑π∈Sn

tbr(π)

are related to the Eulerian polynomials by the identity

P brn (t) =

(1 + t

2

)n−1

(1 + v)n+1An

(1− v1 + v

)(29)

for n ≥ 1, where v =√

1−t1+t

. This identity was proven by David and Barton [9] using

differential equations. Combining (26) and (29), we get the identity

P udrn (t) =

(1 + t

2

)n(1 + v)n+1An

(1− v1 + v

)(30)

expressing the nth udr polynomial

P udrn (t) :=

∑π∈Sn

tudr(π)

67


in terms of the nth Eulerian polynomial for n ≥ 1.

These results have the surprising consequence that the distributions of the statistics pk, lpk,

br, and udr over Sn can all be derived from the distribution of des over Sn (or in the case of

lpk, from the distributions of des over Sk for all k ≤ n).

In this chapter, we greatly expand upon the theme set by the above formulas. We begin

by introducing signed permutations (elements of the hyperoctahedral groups Bn), the notion

of descents in signed permutations, the signed permutation statistics desB, fdes, (desB, neg),

and (fdes, neg), and polynomials that encode the distributions of these signed permutation

statistics over Bn. We prove several preliminary formulas relating these polynomials with

each other and with the Eulerian polynomials.

Our main results use noncommutative symmetric functions to establish several new

identities which relate Eulerian polynomials to polynomials counting permutations by other

descent statistics, including refinements of the known results on pk and lpk proved by

Stembridge and Petersen, respectively. Furthermore, we find expressions for q-exponential

generating functions for q-analogues of these descent statistic polynomials that also keep

track of the inversion number (or inverse major index; see Section 4.5), although there are no

analogous expressions in terms of the q-Eulerian polynomials An(q, t) =∑

π∈Snqinv(π)tdes(π)+1.

In particular, the descent statistics that we consider are the ordered pairs (pk, des) and

(lpk, des), the number of up-down runs udr, and the triple (lpk, val, des). Finally, we relate

the distribution of (lpk, des) over Sn to that of (neg, desB) over Bn, which specializes to

a connection between lpk and desB previously discovered by Petersen [42], and relate the

distribution of (lpk, val, des) over Sn to that of (neg, fdes) over Bn, which specializes to a

previously unknown connection between udr and fdes.

These main results are obtained by applying the homomorphisms Φ and Φq to the

noncommutative symmetric function formulas given in Section 2.3, but it is worth noting that

68


generalizations of several of these results can be proven combinatorially using the modified

Foata–Strehl action and a group action of Petersen on signed permutations; see Section 5 of

[61].

4.2. Descents of type B permutations

The idea of descents has been extended to other finite Coxeter groups, the most important

of which (from the perspective of permutation enumeration) are the hyperoctahedral groups

Bn, consisting of signed permutations. As with Sn, we will not be concerned with the group

structure of Bn, but will study several statistics defined on signed permutations which are

related to type B analogues of descents.

A signed (or type B) n-permutation is a permutation π = π−n · · · π−1π0π1 · · · πn of the set

{−n, . . . ,−1, 0, 1, . . . , n} satisfying π−i = −πi for all −n ≤ i ≤ n. Let Bn be the set of signed

n-permutations. For any signed n-permutation π, we must have π0 = 0 and π is completely

determined by {π1, . . . , πn}, so we can write π as π = π1 · · · πn with the understanding that

π0 = 0 and π−i = −πi for all i. In this way, we can think of Sn as the subset of signed

permutations in Bn with no negative letters among {π1, . . . , πn}.

For cleaner notation, let us write i rather than −i when writing out the letters of a signed

permutation. For example, if π = π1π2π3 with π1 = 3, π2 = −2, and π3 = −1, then we write

π = 321.

We say that i ∈ {0} ∪ [n − 1] is a descent (or type B descent) of π ∈ Bn if πi > πi+1.

Note that we allow 0 to be a descent, which happens precisely when π1 is negative. There

are two notions of descent number for signed permutations that we consider. The descent

number (or type B descent number) desB(π) is simply the number of descents of π ∈ Bn,

whereas the flag descent number fdes(π) is defined by

fdes(π) :=

{2 desB(π), if π1 > 0,

2 desB(π)− 1, if π1 < 0;

69


that is, every descent except 0 is counted twice. For example, let π = 4726351. Then the

descents of π are 0, 2, 3, and 6, so desB(π) = 4 and fdes(π) = 7.

We define

Bn(t) :=∑π∈Bn

tdesB(π)

and

Fn(t) :=∑π∈Bn

tfdes(π),

which are type B analogues of Eulerian polynomials using the descent number and flag descent

number, respectively. We call Bn(t) the nth type B Eulerian polynomial and Fn(t) the nth

flag descent polynomial.

The exponential generating function∞∑n=0

Bn(t)

(1− t)n+1

xn

n!=

ex

1− te2x(31)

was found by Steingrímsson [57], and the analogous formula for the flag descent polynomials∞∑n=0

Fn(t)

(1− t)(1− t2)nxn

n!=

ex

1− tex(32)

directly follows from a result of Adin, Brenti, and Roichman [1, Theorem 4.2].

We consider another statistic on signed permutations: the number of negative letters

neg(π) := |{ πi : πi < 0 and i ∈ [n] }|.

So given π = 4726351, we have neg(π) = 3. We refine the polynomials Bn(t) and Fn(t) by

this statistic, defining

Bn(y, t) :=∑π∈Bn

yneg(π)tdesB(π)

and

Fn(y, t) :=∑π∈Bn

yneg(π)tfdes(π).

70


The polynomials Bn(y, t) were first studied by Brenti [5], but the Fn(y, t) appear to be new.

Later, we will relate the polynomials Bn(y, t) to the joint distribution of lpk and des over

Sn, and similarly Fn(y, t) with the joint distribution of lpk, val, and des. In doing so, we

shall need the following exponential generating functions for these polynomials.

Theorem 4.1.∞∑n=0

Bn(y, t)

(1− t)n+1

xn

n!=

ex

1− te(1+y)x

This is Theorem 3.4 (iv) of [5]. Note that setting y = 1 yields Steingrímsson’s formula

(31).

Proof. We begin by proving the identity

Bn(y, t)

(1− t)n+1=∞∑k=0

(ky + (k + 1))ntk, (33)

which was previously established by Brenti [5, Theorem 3.4 (ii)] and Petersen [42] using

different methods.2

Consider the left-hand side. Each term in Bn(y, t) corresponds to a signed n-permutation

with a vertical bar inserted after each letter corresponding to a descent (and an initial bar if

0 is a descent). For example, if we have π = 4726351, then we write this as

|47|2|635|1.

The 1/(1 − t)n+1 factor corresponds to inserting any number of bars in any of the n + 1

positions between letters, before the first letter, or after the final letter. So for example,

continuing from above, we may have

|47|2||63|||5|1|.2In the paper [61] by the present author, we incorrectly claim that (33) was first proven by Petersen [42].We thank Francesco Brenti for pointing out this error and referring us to his paper [5].

71


Thus the left-hand side of (33) counts the number of signed n-permutations with any number of

bars inserted in any of the n+ 1 positions and at least one bar in every position corresponding

to a descent, where y is weighting the number of negative letters and t is weighting the

number of bars.

We claim that the right-hand side counts these same barred signed n-permutations. Fix

k ≥ 0; this is the number of bars. The bars create k + 1 “boxes” for inserting letters. For

every i ∈ [n], we make two choices: whether or not to make it negative, and which box to

put it into. The letters in each box are then placed in increasing order. Note that the first

box cannot contain any negative letters; otherwise, 0 would be a descent, but there would

not be a bar preceding the first letter. Thus, if a letter is made negative, then it contributes

a weight of y and we can place it in any of the k boxes after the first one. If a letter remains

positive, then it can be placed into any of the k + 1 boxes. Since there are n letters and the

choices are made independently, we have a total contribution of (ky + (k + 1))ntk in the case

where there are k bars in total. Summing over all k yields the right-hand side of (33).

Now, observe that∞∑n=0

Bn(y, t)

(1− t)n+1

xn

n!=∞∑n=0

∞∑k=0

(ky + (k + 1))ntkxn

n!

=∞∑k=0

e(ky+(k+1))xtk

= ex∞∑k=0

(e(1+y)x)ktk

=ex

1− te(1+y)x,

thus completing the proof. �

Theorem 4.2.∞∑n=0

Fn(y, t)

(1− t)(1− t2)nxn

n!=ex + te(1+y)x

1− t2e(1+y)x

72


Setting y = 1 yields the formula (32) of Adin, Brenti, and Roichman.

Proof. We first prove the identity

Fn(y, t)

(1− t)(1− t2)n=∞∑k=0

(ky + (k + 1))nt2k +∞∑k=0

((k + 1)(y + 1))nt2k+1. (34)

Each term in Fn(y, t) corresponds to a signed n-permutation with an arrangement of bars,

but now each nonzero descent contributes a weight of t2, which corresponds to two bars. We

write each π ∈ Bn as π = π−n · · · π−1π1 · · · πn (without π0 = 0). For each descent i ∈ [n− 1],

we insert a bar immediately after πi and a bar immediately before π−i. If i = 0 is a descent,

then we insert a single bar between π−1 and π1. For example, for π = 4726351, we have

1|536|2|74|47|2|635|1.

The 1/(1 − t) factor corresponds to inserting any number of bars in the central position

(between π−1 and π1), and the 1/(1− t2)n factor corresponds to inserting any number of bars

in any of the n positions to the right of the central position, and for each of these bars, a

corresponding bar in the position symmetric about the center. For example, we may have

|1|5|||36||2|74|47|2||63|||5|1|.

We claim that the right-hand side of (34) counts the same arrangements. We consider

two cases: the number of bars is even or the number of bars is odd.

• Suppose that the number of bars is 2k for some k ≥ 0. These bars create 2k + 1

boxes; we will be inserting letters into the right-most k + 1 boxes. Again, for each

letter, we decide whether or not to make it negative and decide which box to put it

in. If a letter is made negative, then it contributes a weight of y and it can only be

inserted into the final k boxes; if a letter is not made negative, then we can insert it

into any of the right-most k + 1 boxes. Order the letters in each box in increasing

73


order, and for each letter, insert its negative into the position symmetric about the

center. These are precisely the arrangements that we want; thus we have the term

(ky + (k + 1))nt2k, and summing over k gives the total contribution from having an

even number of bars.

• Suppose that the number of bars is 2k + 1 for some k ≥ 0. Then these bars create

2k + 2 boxes. In this case, both positive and negative letters can be inserted into

any of the right-most k + 1 boxes; if a negative letter is inserted in the (k + 2)nd

box, then the central bar acts as the bar corresponding to the 0 descent. Hence, this

contributes ((k + 1)(y + 1))nt2k, and summing over k gives the total contribution

from having an odd number of bars.

Now, observe that∞∑n=0

∞∑k=0

(ky + (k + 1))nt2kxn

n!=∞∑k=0

e(k(1+y)+1)xt2k

= ex∞∑k=0

(t2e(1+y)x)k

=ex

1− t2e(1+y)x

and∞∑n=0

∞∑k=0

((k + 1)(y + 1))nt2k+1xn

n!=∞∑k=0

e((k+1)(y+1))xt2k+1

= te(1+y)x

∞∑k=0

(t2e(1+y)x)k

=te(1+y)x

1− t2e(1+y)x;

adding these expressions completes the proof. �

74


4.3. Several new Eulerian polynomial identities

Before proceeding, we prove several new formulas relating the Eulerian polynomials An(t),

refined type B Eulerian polynomials Bn(y, t), and refined flag descent polynomials Fn(y, t).

These results are of a similar flavor to some of the main results of Chapter 4, but can be

obtained simply using the exponential generating functions established in Section 4.2 and do

not require the use of noncommutative symmetric functions.

Theorem 4.3. For n ≥ 0, we have

Bn(y, t) =n∑k=0

(n

k

)(1 + y)k(1− t)n−kAk(t).

Proof. Taking Theorem 4.1, multiplying both sides by 1− t, and then replacing x with

(1− t)x/(1 + y) yields∞∑n=0

Bn(y, t)

(1 + y)nxn

n!=

1− t1− te(1−t)x e

1−t1+y

x

=( ∞∑n=0

An(t)xn

n!

)( ∞∑n=0

(1− t1 + y

)nxn

n!

)=∞∑n=0

n∑k=0

(n

k

)(1− t1 + y

)n−kAk(t)

xn

n!.

Equating the coefficients of xn/n! and multiplying both sides by (1 +y)n yields the result. �

By setting y = 1, we obtain the following corollary.

Corollary 4.4. For n ≥ 0, we have

Bn(t) =n∑k=0

(n

k

)2k(1− t)n−kAk(t).


Fn(y, t) =1

1 + t

((1 + y)n

tAn(t2) +

n∑k=0

(n

k

)(1 + y)k(1− t2)n−kAk(t

2)).

75


Proof. It is readily checked that the statement of Theorem 4.2 is equivalent to

1

1− t

(1 + t

∞∑n=1

Fn(y, t)

(1− t2)nxn

n!

)=

1 + tex

1− t2e(1+y)x.

Multiplying both sides by 1− t2 and replacing x with (1− t2)x/(1 + y) yields

(1 + t)(

1 + t∞∑n=1

Fn(y, t)

(1 + y)nxn

n!

)=

1− t2

1− t2e(1−t2)x(1 + te

1−t2

1+yx)

=( ∞∑n=0

An(t2)xn

n!

)(1 + t

∞∑n=0

(1− t2

1 + y

)nxn

n!

)=∞∑n=0

(An(t2) + t

n∑k=0

(n

k

)(1− t2

1 + y

)n−kAk(t

2))xnn!.

Equating the coefficients of xn/n! and dividing both sides by t(1 + t)/(1 + y)n yields the

result. �

We can set y = 1 to obtain an identity relating Fn(t) and An(t), but the nicer identity

Fn(t) =(1 + t)n

tAn(t) (35)

can be obtained by directly comparing the generating functions of Fn(t) and An(t), and can

also be recovered as a specialization of a more general identity of Adin, Brenti, and Roichman

[1, Theorem 4.4].


Fn(y, t) =1

1 + t

(Bn(y, t2) +

1

t

n∑k=0

(−1)n−k(n

k

)(1− t2)n−kBk(y, t

2)).

Proof. Due to Theorem 4.5, it suffices to show that

Bn(y, t2) =n∑k=0

(n

k

)(1 + y)k(1− t2)n−kAk(t

2)

and that(1 + y)n

tAn(t2) =

1

t

n∑k=0

(−1)n−k(n

k

)(1− t2)n−kBk(y, t

2).

76


Note that Theorem 4.3 directly implies the former equation, whereas a simple application of

inclusion-exclusion to Theorem 4.3 implies the latter equation. �

Rather than stating the result of setting y = 1 in Theorem 4.6, we give a simpler identity

relating the polynomials Fn(t) and Bn(t) using (35).


Fn(t) =1

t

(1 + t

2

)n n∑k=0

(−1)n−k(n

k

)(1− t)n−kBk(t).

Proof. By applying inclusion-exclusion to Corollary 4.4, we obtain

An(t) =1

2n

n∑k=0

(−1)n−k(n

k

)(1− t)n−kBk(t) (36)

(for n ≥ 1). Combining (35) and (36) yields the result. �

4.4. Main results

4.4.1. On peaks and descents. Consider the polynomial P (pk,des)n (y, t) defined by

P (pk,des)n (y, t) :=

∑π∈Sn

ypk(π)+1tdes(π)+1

for n ≥ 1 and by P (pk,des)0 (y, t) := 1, which refines the Eulerian polynomial An(t) and the

pk polynomial P pkn (t). We prove in our first theorem an identity expressing P (pk,des)

n (y, t) in

terms of An(t).


An(t) =

(1 + yt

1 + y

)n+1

P (pk,des)n

((1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

). (37)

Equivalently,

P (pk,des)n (y, t) =

(1 + u

1 + uv

)n+1

An(v) (38)

where u =1+t2−2yt−(1−t)

√(1+t)2−4yt

2(1−y)tand v =

(1+t)2−2yt−(1+t)√

(1+t)2−4yt

2yt.

77


Note that evaluating (37) at y = 1 recovers Stembridge’s identity (27).

Proof. Taking Lemma 2.5, evaluating at x = 1, and applying the homomorphism Φ

yields

1

1− te(1+y)x=

1

1− t+∞∑n=1

∑π∈Sn

tpk(π)+1(y + t)des(π)−pk(π)(1 + yt)n−pk(π)−des(π)−1(1 + y)2 pk(π)+1

(1− t)n+1

xn

n!

by Lemma 2.1. Rearranging some terms yields

1

1− te(1+y)x=

1

1− t+∞∑n=1

∑π∈Sn

1

1 + y

(1 + yt

1− t

)n+1((1 + y)2t

(y + t)(1 + yt)

)pk(π)+1(y + t

1 + yt

)des(π)+1xn

n!.

Multiplying both sides by 1− t and then replacing x by (1− t)x/(1 + y) yields

1− t1− te(1−t)x = 1 +

∞∑n=1

∑π∈Sn

(1 + yt

1 + y

)n+1((1 + y)2t

(y + t)(1 + yt)

)pk(π)+1(y + t

1 + yt

)des(π)+1xn

n!.

Note that the left-hand side is the exponential generating function for the Eulerian polynomials;

thus equating the coefficients of xn/n! gives (37).

Finally, (38) can be obtained by setting u = (1+y)2t(y+t)(1+yt)

and v = y+t1+yt

, solving for y and t

(which can be done using Maple), and simplifying.3 �

Surprisingly, the left-hand side of (37) does not depend on y, indicating that all terms on

the right-hand side involving y cancel out.

Next, we obtain a formula for the q-exponential generating function for the q-analogue of

the (pk, des) polynomial also keeping track of the inversion number. Define P (inv,pk,des)n (q, y, t)

3We exchanged u and v with y and t, respectively, in the statement of (38) in this theorem, so that the(pk,des) polynomial would have variables y and t as in its definition. The same is done for all subsequentresults involving similar substitutions.

78


by

P (inv,pk,des)n (q, y, t) :=

∑π∈Sn

qinv(π)ypk(π)+1tdes(π)+1

for n ≥ 1 and by P (inv,pk,des)0 (q, y, t) := 1.

Theorem 4.9. We have

1− t1− tExpq(yx) expq(x)

=

1 +∞∑n=1

(1 + yt)n+1

(1 + y)(1− t)nP (inv,pk,des)n

(q,

(1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

)xn

[n]q!. (39)

Equivalently,

∞∑n=1

P (inv,pk,des)n (q, y, t)

xn

[n]q!=v(1 + u)

1 + uv

Expq

(u(1−v)1+uv

x)

expq(

1−v1+uv

x)− 1

1− v Expq

(u(1−v)1+uv

x)

expq(

1−v1+uv

x)


√(1+t)2−4yt

2(1−y)tand v =

(1+t)2−2yt−(1+t)√

(1+t)2−4yt

2yt.

Proof. We follow the proof of Theorem 4.8, but apply the homomorphism Φq instead of

Φ, which by Lemma 2.3 yields

1

1− tExpq(yx) expq(x)=

1

1− t

+∞∑n=1

∑π∈Sn

qinv(π) tpk(π)+1(y + t)des(π)−pk(π)(1 + yt)n−pk(π)−des(π)−1(1 + y)2 pk(π)+1

(1− t)n+1

xn

[n]q!.

Multiplying both sides by 1− t and rearranging some terms yields (39).

Next, we replace x by (1− t)x/(1 + yt) to get

1− t

1− tExpq

(y(1−t)1+yt

x)

expq

(1−t1+yt

x) =

1 +1 + yt

1 + y

∞∑n=1

P (inv,pk,des)n

(q,

(1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

)xn

[n]q!.

79


Making the same substitutions yields

1 +1 + uv

1 + u

∞∑n=1

P (inv,pk,des)n (q, y, t)

xn

[n]q!=

1− v

1− v Expq

(u(1−v)1+uv

x)

expq(

1−v1+uv

x)


√(1+t)2−4yt

2(1−y)tand v =

(1+t)2−2yt−(1+t)√

(1+t)2−4yt

2yt. Subtracting both sides

by 1 and dividing by (1 + uv)/(1 + u) completes the proof. �

Unfortunately, we cannot express P (inv,pk,des)n (q, y, t) in terms of the q-Eulerian polynomial

An(q, t) defined by

An(q, t) :=∑π∈Sn

qinv(π)tdes(π)+1

for n ≥ 1 and by A0(q, t) := 1, but we can recover the known q-exponential generating

function for the q-Eulerian polynomials from the above result.

Corollary 4.10.∞∑n=0

An(q, t)xn

[n]q!=

1− t1− t expq((1− t)x)

Proof. Take (39), set y = 0, and replace x with (1− t)x. �

Theorem 4.9 also specializes to a corresponding result for the (inv, pk) polynomial

P(inv,pk)n (q, t) defined by

P (inv,pk)n (q, t) :=

∑π∈Sn

qinv(π)tpk(π)+1

for n ≥ 1 and by P (inv,pk)0 (q, t) := 1.

Corollary 4.11. We have

1− t1− tExpq(x) expq(x)

= 1 +∞∑n=1

(1 + t)n+1

2(1− t)nP (inv,pk)n

(q,

4t

(1 + t)2

)xn

[n]q!. (40)

Equivalently,∞∑n=1

P (inv,pk)n (q, t)

xn

[n]q!=

2v

1 + v

Expq(

1−v1+v

x)

expq(

1−v1+v

x)− 1

1− v Expq(

1−v1+v

x)

expq(

1−v1+v

x) (41)

80


where v = 2t(1−

√1− t)− 1.

Proof. Equation (40) is obtained by taking (39) and setting y = 1. Then (41) follows

by replacing x with x(1− t)/(1 + t), making an appropriate substitution, and rearranging

some terms. �

4.4.2. On left peaks and descents. In this section, we study the (lpk, des) polynomials

P (lpk,des)n (y, t) :=

∑π∈Sn

ylpk(π)tdes(π)

and their q-analogues

P (inv,lpk,des)n (q, y, t) :=

∑π∈Sn

qinv(π)ylpk(π)tdes(π).

Using the same method as in the previous section, we obtain analogues of Theorems 4.8

and 4.9 for left peaks and descents, as well as a connection to the refined type B Eulerian

polynomials introduced in Section 4.3.

Theorem 4.12. For n ≥ 0, we haven∑k=0

(n

k

)(1 + y)k(1− t)n−kAk(t) = (1 + yt)nP (lpk,des)

n

((1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

). (42)

Equivalently,

P (lpk,des)n (y, t) =

1

(1 + uv)n

n∑k=0

(n

k

)(1 + u)k(1− v)n−kAk(v) (43)


√(1+t)2−4yt

2(1−y)tand v =

(1+t)2−2yt−(1+t)√

(1+t)2−4yt

2yt.

As with Theorem 4.8, evaluating (42) at y = 1 recovers a known result, Petersen’s identity

(28) in this case.

81


Proof. Taking Lemma 2.6, evaluating at x = 1, applying the homomorphism Φ, and

rearranging some terms yields

ex

1− te(1+y)x=

1

1− t+∞∑n=1

∑π∈Sn

(1 + yt)n

(1− t)n+1

((1 + y)2t

(y + t)(1 + yt)

)lpk(π)(y + t

1 + yt

)des(π)xn

n!.

Multiplying both sides by 1− t and then replacing x by (1− t)x/(1 + y) yields

1− t1− te(1−t)x e

1−t1+y

x =∞∑n=0

∑π∈Sn

(1 + yt

1 + y

)n((1 + y)2t

(y + t)(1 + yt)

)lpk(π)(y + t

1 + yt

)des(π)xn

n!

Moreover,

1− t1− te(1−t)x e

1−t1+y

x =( ∞∑n=0

An(t)xn

n!

)( ∞∑n=0

(1− t1 + y

)nxn

n!

)=∞∑n=0

n∑k=0

(n

k

)Ak(t)

(1− t1 + y

)n−kxn

n!,

so

∞∑n=0

n∑k=0

(n

k

)Ak(t)

(1− t1 + y

)n−kxn

n!=

∞∑n=0

∑π∈Sn

(1 + yt

1 + y

)n((1 + y)2t

(y + t)(1 + yt)

)lpk(π)(y + t

1 + yt

)des(π)xn

n!.

Equating the coefficients of xn/n! and rearranging some terms gives (42). Then (43) can be

obtained by making the same substitutions as in the proof of Theorem 4.8. �

Now, for the q-analogue.


(1− t) expq(x)

1− tExpq(yx) expq(x)=∞∑n=0

(1 + yt

1− t

)nP (inv,lpk,des)n

(q,

(1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

)xn

[n]q!. (44)


P (inv,lpk,des)n (q, y, t)

xn

[n]q!=

(1− v) expq(

1−v1+uv

x)

1− v Expq

(u(1−v)1+uv

x)

expq(

1−v1+uv

x) (45)

82



√(1+t)2−4yt

2(1−y)tand v =

(1+t)2−2yt−(1+t)√

(1+t)2−4yt

2yt.

Proof. Apply the homomorphism Φq to Lemma 2.6 evaluated at x = 1; then multiplying

both sides by 1− t yields (44).

Next, replace x by (1− t)x/(1 + yt) to get

(1− t) expq

(1−t1+yt

x)

1− tExpq

(y(1−t)1+yt

x)

expq

(1−t1+yt

x) =

∞∑n=0

P (inv,lpk,des)n

(q,

(1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

)xn

[n]q!.

Making the same substitutions as before yields (45). �

We note that taking (44), evaluating at y = 0, and substituting (1− t)x for x gives

(1− t) expq((1− t)x)

1− t expq((1− t)x)=∞∑n=0

∑π∈Sn

qinv(π)tdes(π) xn

[n]q!,

which is equivalent to Corollary 4.10. Evaluating at y = 1, on the other hand, gives us a

result for the (inv, lpk) polynomials

P (inv,lpk)n (q, t) :=

∑π∈Sn

qinv(π)tlpk(π).

Corollary 4.14. We have

(1− t) expq(x)

1− tExpq(x) expq(x)=∞∑n=0

(1 + t

1− t

)nP (inv,lpk)n

(q,

4t

(1 + t)2

)xn

[n]q!.


P (inv,lpk)n (q, t)

xn

[n]q!=

(1− v) expq(

1−v1+v

x)

1− v Expq(

1−v1+v

x)

expq(

1−v1+v

x)

where v = 2t(1−

√1− t)− 1.

Lastly, we state an identity connecting the (lpk, des) polynomials with the refined type B

Eulerian polynomials Bn(y, t) =∑

π∈Bnyneg(π)tdesB(π).

83


Theorem 4.15. For all n ≥ 0, we have

Bn(y, t) = (1 + yt)nP (lpk,des)n

((1 + y)2t

(y + t)(1 + yt),y + t

1 + yt

). (46)

Equivalently,

P (lpk,des)n (y, t) =

Bn(u, v)

(1 + uv)n(47)


√(1+t)2−4yt

2(1−y)tand v =

(1+t)2−2yt−(1+t)√

(1+t)2−4yt

2yt.

By setting y = 1 in (46), we recover the identity

Bn(t) = (1 + t)nP lpkn

(4t

(1 + t)2

),

which is another result of Petersen [42].

Proof. Observe that (46) follows immediately from Theorems 4.3 and 4.12. Making the

same substitutions as before yields (47). �

4.4.3. On up-down runs and descents. Our remaining aim in this chapter is to prove

analogous results for the number of up-down runs udr and the joint distribution of udr and des.

In particular, we rederive (30) in an equivalent form, give a q-exponential generating function

for the q-analogue of the polynomial P udrn (y, t), and relate P udr

n (y, t) to the distribution of

the flag descent number over Bn. Due to technical constraints that will become apparent

later, we cannot do the same with the polynomial

P (udr,des)n (y, t) :=

∑π∈Sn

yudr(π)tdes(π).

Instead, we will work with

P (lpk,val,des)n (y, z, t) :=

∑π∈Sn

ylpk(π)zval(π)tdes(π),

which is equivalent to P (udr,des)n (y, t) in light of Lemma 1.4.

84



An(t) =(1 + t2)n

2(1 + t)n−1P udrn

(2t

1 + t2

). (48)

Equivalently,

P udrn (t) =

2(1 + v)n−1

(1 + v2)nAn(v) (49)

where v = 1−√

1−t2t

.

Proof. Taking Corollary 2.8, evaluating at x = 1, applying the homomorphism Φ, and

rearranging some terms yields

1

1− tex=

1 + tex

1− t2e2x=

1

1− t+∞∑n=1

∑π∈Sn

(1 + t2)n

2(1− t)2(1− t2)n−1

(2t

1 + t2

)udr(π)xn

n!.

Multiplying both sides by 1− t and then replacing x by (1− t)x yields

1− t1− te(1−t)x = 1 +

∞∑n=1

∑π∈Sn

(1 + t2)n

2(1 + t)n−1

(2t

1 + t2

)udr(π)xn

n!.

The left-hand side is precisely the exponential generating function for the Eulerian polynomials,

so equating the coefficients of xn/n! gives (48). Then (49) can be obtained by making the

substitution v = 2t/(1 + t2) and solving for t. �

One may verify that (49) is equivalent to the identity (30) derived from the identities of

David and Barton (29) and Stanley (26).

Now, define

P (inv,udr)n (q, t) :=

∑π∈Sn

qinv(π)tudr(π).


(1− t)(1 + t expq(x))

1− t2 expq(x) Expq(x)= 1 +

1 + t

2

∞∑n=1

(1 + t2

1− t2

)nP (inv,udr)n

(q,

2t

1 + t2

)xn

[n]q!. (50)

85


Equivalently,

∞∑n=1

P (inv,udr)n (q, t)

xn

[n]q!=

2

1 + v

(1− v)(

1 + v expq

(1−v21+v2

x))

1− v2 expq(

1−v21+v2

x)

Expq(

1−v21+v2

x) − 1

(51)

where v = 1−√

1−t2t

.

Proof. Apply the homomorphism Φq to Corollary 2.8 evaluated at x = 1; then multiply-

ing both sides by 1− t yields (50).

Next, replace x by x(1− t2)/(1 + t2) to get

(1− t)(

1 + t expq

(1−t21+t2

x))

1− t2 expq(

1−t21+t2

x)

Expq(

1−t21+t2

) = 1 +1 + t

2

∞∑n=1

P (inv,udr)n

(q,

2t

1 + t2

)xn

[n]q!.

Then rearranging some terms and making the same substitution as in the proof of Theorem

4.16 yields (51). �

Recall that when setting y = 1 in Lemma 2.7, all instances of the statistics lpk and val

either cancel out or reduce to udr. This is not possible in the general form of Lemma 2.7,

so we cannot directly work with the polynomials P (udr,des)n (y, t) =

∑π∈Sn

yudr(π)tdes(π). Since

the statistics (udr, des) and (lpk, val, des) are equivalent, we will instead give results for the

polynomials P (lpk,val,des)n (y, z, t) =

∑π∈Sn

ylpk(π)zval(π)tdes(π) and their q-analogues

P (inv,lpk,val,des)n (q, y, z, t) :=

∑π∈Sn

qinv(π)ylpk(π)zval(π)tdes(π).


(1 + y)n

tAn(t2) +

n∑k=0

(n

k

)(1 + y)k(1− t2)n−kAk(t

2) = (1 + yt)(1 + t)(1 + yt2)n−1

× P (lpk,val,des)n

(t(1 + y)(y + t)

(y + t2)(1 + yt),t(1 + y)(1 + yt)

(1 + yt2)(y + t),y + t2

1 + yt2

).

86


Theorem 4.19.

(1− t)(1 + t expq(x))

1− t2 expq(x) Expq(yx)= 1 + t(1 + yt)

∞∑n=1

(1 + yt2)n−1

(1− t2)n

× P (inv,lpk,val,des)n

(q,t(1 + y)(y + t)

(y + t2)(1 + yt),t(1 + y)(1 + yt)

(1 + yt2)(y + t),y + t2

1 + yt2

)xn

[n]q!

We omit the proofs of the above two theorems as they follow in essentially the same way

as the proofs of Theorems 4.16 and 4.17, except that we would use Lemma 2.7 rather than its

specialization (Corollary 2.8). Unlike in these theorems, however, it is not possible to invert

the identities to give an explicit expression for P (lpk,val,des)n (y, z, t) or for the q-exponential

generating function for P (inv,lpk,val,des)n (q, y, z, t).

Finally, we relate the (lpk, val, des) polynomials to the refined flag descent polynomials

Fn(y, t) =∑

π∈Bnyneg(π)tfdes(π), which specializes to a relation between the udr polynomials

and the flag descent polynomials Fn(t) =∑

π∈Bntfdes(π).


Fn(y, t) = (1 + yt)(1 + yt2)n−1P (lpk,val,des)n

(t(1 + y)(y + t)

(y + t2)(1 + yt),t(1 + y)(1 + yt)

(1 + yt2)(y + t),y + t2

1 + yt2

).

Proof. Follows immediately from Theorems 4.5 and 4.18. �


Fn(t) =(1 + t)(1 + t2)n

2tP udrn

(2t

1 + t2

). (52)

Equivalently,

P udrn (t) =

2v

(1 + v)(1 + v2)nFn(v) (53)

where v = 1−√

1−t2t

.

Proof. We obtain (52) by taking the preceding theorem, setting y = 1, and rearranging

a few terms. Then (53) is obtained from (52) by the same substitution as before. �

87


4.5. Two remarks: the inverse major index and alternating analogues

We end this chapter with two brief remarks. First, our formulas for q-analogues of descent

statistic polynomials are also valid for the “inverse major index”. For a permutation π, the

inverse major index imaj(π) is the major index of its inverse when considered as an element

of the symmetric group. For example, take π = 85712643, whose inverse is π−1 = 45872631.

Since Des(π−1) = {3, 4, 6, 7}, the inverse major index of π is

imaj(π) = maj(π−1) = 3 + 4 + 6 + 7 = 20.

A remarkable result by Foata and Schützenberger [16] states that the inversion number

inv and inverse major index imaj are equidistributed over descent classes. That is, for any

S ⊆ [n− 1], ∑π∈Sn

Des(π)=S

qinv(π) =∑π∈Sn

Des(π)=S

qimaj(π);

this is equivalent to saying that the polynomial βq(L)—defined in the statement of Lemma

1.2—counting n-permutations with descent composition L by inversion number also counts

these same permutations by inverse major index. It follows that Theorems 4.9, 4.13, 4.17,

and 4.19 and Corollaries 4.11 and 4.14 can be restated for the inverse major index, that is,

by replacing every instance of inv with imaj.

Second, although we do not present them here, we mention that there exist alternating

analogues (in the sense of Sections 1.4 and 2.2) of Theorems 4.8, 4.12, 4.16, and 4.18 that

express alternating analogues of descent statistic polynomials in terms of the alternating

Eulerian polynomials. For example, since i is a peak of π precisely when i− 1 is an ascent

and i is a descent, we define i to be an alternating peak if i − 1 is an “alternating ascent”

and i is an alternating descent, i.e., if πi−1 > πi > πi+1 and i is odd or if πi−1 < πi < πi+1

and i is even. Let altpk(π) be the number of alternating peaks of π. Then, by applying Φ

instead of Φ in the proof of Theorem 4.8, we would obtain an expression for the polynomial

88


P(altpk,altdes)n (y, t) :=

∑π∈Sn

yaltpk(π)+1taltdes(π)+1 in terms of the nth alternating Eulerian

polynomial.

89

CHAPTER 5

Shuffle-compatible permutation statistics

5.1. Introduction

For this chapter, we redefine a permutation of length n (or an n-permutation) to be a

sequence of n distinct letters—not necessarily from 1 to n—in P. For example, π = 47381 is

a permutation of length 5. Let Pn denote the set of all n-permutations in this sense.

Define the standardization of π ∈ Pn to be the permutation in Sn obtained by replacing

the ith smallest letter of π with i for every i from 1 to n. For example, the standardization

of 47381 is 34251. We say that two permutations are order-isomorphic if they have the same

standardization. It is clear that order-isomorphism is an equivalence relation and that its

equivalence classes can be identified with permutations in Sn.

We also redefine a permutation statistic st to be a function defined on permutations such

that st(π) = st(σ) whenever π and σ are order-isomorphic. Every permutation statistic that

has been defined so far is a permutation statistic in this sense.

Let π ∈ Pm and σ ∈ Pn be disjoint permutations, that is, permutations with no

letters in common. We say that τ ∈ Pm+n is a shuffle of π and σ if both π and σ are

subsequences of τ . The set of shuffles of π and σ is denoted S(π, σ). For example, S(53, 16) =

{5316, 5136, 5163, 1653, 1536, 1563}. It is easy to see that the number of permutations in

S(π, σ) is(m+nm

).

Richard Stanley’s theory of P -partitions [51] implies that the descent set statistic has

a remarkable property related to shuffles: for disjoint permutations π and σ, the multiset

{Des(τ) : τ ∈ S(π, σ) }—that is, the distribution of the descent set over shuffles of π and

90

CHAPTER 5. SHUFFLE-COMPATIBLE PERMUTATION STATISTICS

σ—depends only on Des(π), Des(σ), and the lengths of π and σ [55, Exercise 3.161]. That

is, if π and π′ are permutations of the same length with the same descent set, and similarly

with σ and σ′, then the number of permutations in S(π, σ) with any given descent set is the

same as the number of permutations in S(π′, σ′) with that descent set.

Stanley also proved a similar but more refined result for the joint statistic (des,maj),

which is a special case of [51, Proposition 12.6 (ii)]. Bijective proofs were later found by

Goulden [25] and by Stadler [50]; they referred to this result as “Stanley’s shuffling theorem”.

Theorem 5.1 (Stanley’s shuffling theorem). Let π ∈ Pm and σ ∈ Pn be disjoint

permutations, and let Sk(π, σ) be the set of shuffles of π and σ with exactly k descents. Then∑τ∈Sk(π,σ)

qmaj(τ) = qmaj(π)+maj(σ)+(k−des(π))(k−des(σ))

×(m− des(π) + des(σ)

k − des(π)

)q

(n− des(σ) + des(π)

k − des(σ)

)q

. (54)

A variant of the theorem gives the formula∑τ∈S(π,σ)

qmaj(τ) = qmaj(π)+maj(σ)

(m+ n

m

)q

; (55)

see [51, p. 43]. These formulas show that the statistics (des,maj) and maj have the same

property as Des, and setting q = 1 in (54) shows that des has this property as well.

We call this property “shuffle-compatibility”. More precisely, we say that a permutation

statistic st is shuffle-compatible if for disjoint permutations π and σ, the distribution of st

over S(π, σ) depends only on st(π), st(σ), |π|, and |σ|. Hence Des, des, maj, and (des,maj)

are examples of shuffle-compatible permutation statistics.

This chapter is a summary of the first in-depth investigation of shuffle-compatibility.

We begin by defining the “shuffle algebra” of a shuffle-compatible permutation statistic

st, which has a natural basis whose structure constants encode the distribution of st over

shuffles of permutations (or more precisely, equivalence classes of permutations induced91


by the statistic st). We then prove several basic results that relate the shuffle algebras of

permutation statistics that are related in various ways; notably, if two statistics are related

by a basic symmetry—reversion, complementation, or reverse complementation—and one of

them is known to be shuffle-compatible, then both statistics are shuffle-compatible and have

isomorphic shuffle algebras.

Next, we develop a framework for studying shuffle-compatible descent statistics. The cor-

nerstone of this framework is a necessary and sufficient condition for the shuffle-compatibility

of a descent statistic, which shows that the shuffle algebra of any shuffle-compatible descent

statistic is isomorphic to a quotient of the algebra QSym of quasisymmetric functions. Then,

we exploit the duality between the algebra structure of QSym and the coalgebra structure of

Sym to obtain a dual version of our shuffle-compatibility condition, which allows us to prove

that a descent statistic is shuffle-compatible by constructing a suitable subcoalgebra of Sym.

We use this machinery to give explicit descriptions of the shuffle algebras of pk, (pk, des),

lpk, (lpk, des), udr, (udr, des), des, and (des,maj), thus showing that these statistics are all

shuffle-compatible.

5.2. Shuffle algebras

5.2.1. Definition and basic results. Every permutation statistic st induces an equiv-

alence relation on permutations; we say that permutations π and σ are st-equivalent if

st(π) = st(σ) and |π| = |σ|.1 We write the st-equivalence class of π as [π]st. For a shuffle-

compatible statistic st, we can then associate to st a Q-algebra in the following way. First,

associate to st a Q-vector space by taking as a basis the st-equivalence classes of permutations.

We give this vector space a multiplication by taking

[π]st[σ]st =∑

τ∈S(π,σ)

[τ ]st,

1The notion of st-equivalence should not be confused with that of “st-Wilf equivalence” [10].

92


which is well-defined (i.e., the choice of π and σ in an equivalence class does not matter)

because st is shuffle-compatible. Conversely, if such a multiplication is well-defined, then st is

shuffle-compatible. We denote the resulting algebra by Ast and call it the shuffle algebra of

st. Observe that Ast is graded, and [π]st belongs to the nth homogeneous component of Ast

if π has length n.

As an example, we describe the shuffle algebra of the major index maj.

Theorem 5.2 (Shuffle-compatibility of the major index).

(a) The major index maj is shuffle-compatible.

(b) The linear map on Amaj defined by

[π]maj 7→qmaj(π)

[|π|]q!x|π|

is a Q-algebra isomorphism from Amaj to the span of{qj

[n]q!xn}n≥0, 0≤j≤(n

2),

a subalgebra of Q[[q]][x].

(c) The nth homogeneous component of Amaj has dimension(n2

)+ 1.

Proof. We know from (55) that maj is shuffle-compatible, so there is no need to prove

(a). Let φ : Amaj → Q[[q]][x] denote the map given in the statement of (b). Then by (55), for

disjoint π ∈ Pm and σ ∈ Pn, we have

φ([π]maj)φ([σ]maj) =qmaj(π)

[m]q!xm

qmaj(σ)

[n]q!xn

=qmaj(π)+maj(σ)

[m]q![n]q!xm+n

=qmaj(π)+maj(σ)

[m+ n]q!

(m+ n

m

)q

xm+n

93


=∑

τ∈S(π,σ)

qmaj(τ)

[m+ n]q!xm+n

= φ([π]maj[σ]maj),

so φ is an algebra homomorphism. The possible values for maj(π) for an n-permutation π

range from 0 to(n2

), and since the elements qjxn/[n]q! are linearly independent, φ gives an

isomorphism from Amaj to the stated subalgebra, thus proving (b) and (c). �

We say that two permutation statistics st1 and st2 are equivalent if [π]st1 = [π]st2 for every

permutation π. In other words, st2(π) depends only on st1(π) and |π| for every permutation

π, and vice versa. As shown in Lemma 1.4, udr and (lpk, val) are equivalent statistics.

Theorem 5.3. Suppose that st1 and st2 are equivalent statistics. If st1 is shuffle-compatible

with shuffle algebra Ast1, then st2 is also shuffle-compatible with shuffle algebra Ast2 isomorphic

to Ast1.

Proof. Equivalent statistics have the same equivalence classes on permutations, so Ast1

and Ast2 (as vector spaces) have the same basis elements. If st1 and st2 are equivalent, then

[π]st2 [σ]st2 = [π]st1 [σ]st1 =∑

τ∈S(π,σ)

[τ ]st1 =∑

τ∈S(π,σ)

[τ ]st2 ,

which proves the result. �

We say that st1 is a refinement of st2 if for all permutations π and σ of the same length,

st1(π) = st1(σ) implies st2(π) = st2(σ). For example, the statistics of which the descent set

is a refinement are exactly what we call descent statistics.

Theorem 5.4. Suppose that st1 is shuffle-compatible and is a refinement of st2. Let A be

a Q-algebra with basis {uα} indexed by st2-equivalence classes α, and suppose that there exists

a Q-algebra homomorphism φ : Ast1 → A such that for every st1-equivalence class β, we have

94


φ(β) = uα where α is the st2-equivalence class containing β. Then st2 is shuffle-compatible

and the map uα 7→ α extends by linearity to an isomorphism from A to Ast2.

Proof. It is sufficient to show that for any two disjoint permutations π and σ, we have

u[π]st2u[σ]st2

=∑

τ∈S(π,σ)

u[τ ]st2.

To see this, we have

u[π]st2u[σ]st2

= φ([π]st1)φ([σ]st1)

= φ([π]st1 [σ]st1)

= φ( ∑τ∈S(π,σ)

[τ ]st1

)=

∑τ∈S(π,σ)

u[τ ]st2. �

5.2.2. Basic symmetries yield isomorphic shuffle algebras. Here we consider the

three involutions reversion, complementation, and reverse-complementation and their impli-

cations for the shuffle-compatibility of permutation statistics.

Let f be an involution on the set of permutations which preserves the length of a

permutation. Then let πf denote f(π). Given a set X of permutations, let

Xf := { πf : π ∈ X },

so f naturally induces an involution on sets of permutations as well.

We say that two permutation statistics st1 and st2 are f -equivalent if st1 ◦f is equivalent

to st2. Equivalently, st1 and st2 are f -equivalent if ([πf ]st1)f = [π]st2 for all π. It is easy to

verify that st1(πf ) = st2(π) implies that st1 and st2 are f -equivalent (although this is not a

necessary condition).

95


For example, Lpk and Rpk are r-equivalent, pk and val are c-equivalent, Pk and Val are

c-equivalent, and (pk, des) and (val, des) are rc-equivalent. It is less obvious that (lpk, val)

and (lpk, pk) are rc-equivalent, so we provide a proof below.

Proposition 5.5. (lpk, val) and (lpk, pk) are rc-equivalent statistics.

Proof. Fix a permutation π. We divide into four cases: (a) π has a short initial run and

a long final run, (b) π has a short initial run and a short final run, (c) π has a long initial

run and a long final run, and (d) π has a long initial run and short final run. In case (a),

we know from Lemma 1.4 that lpk(π) = val(π). Then pk(πrc) = val(π), and πrc has a long

initial run, so

lpk(πrc) = pk(πrc) = val(π) = lpk(π).

Thus, (lpk, val)(π) = (lpk, pk)(πrc). The other three cases can be verified in the same way. �

Let us say that f is shuffle-compatibility-preserving if for every pair of disjoint permutations

π and σ, there exist disjoint permutations π and σ with the same relative order as π and σ,

respectively, such that S(πf , σf ) = S(π, σ)f and S(πf , σf ) = S(π, σ)f .

We note that f -equivalences are not actually equivalence relations on statistics (although

they are symmetric), but we shall show that if the statistics are shuffle-compatible and f is

shuffle-compatibility-preserving, then f -equivalences induce isomorphisms on the correspond-

ing shuffle algebras.

Theorem 5.6. Let f be shuffle-compatibility-preserving, and suppose that st1 and st2 are

f -equivalent statistics. If st1 is shuffle-compatible with shuffle algebra Ast1, then st2 is also

shuffle-compatible with shuffle algebra Ast2 isomorphic to Ast1.

Proof. Let π and π be permutations in the same st2-equivalence class and similarly

with σ and σ, such that π and σ are disjoint and π and σ are disjoint. Since st1 and st2 are

96


f -equivalent, it follows that

([πf ]st1)f = [π]st2 = [π]st2 = ([πf ]st1)

f .

Hence [πf ]st1 = [πf ]st1 and similarly [σf ]st1 = [σf ]st1 .

Since f is shuffle-compatibility-preserving, there exist permutations π, σ, ˆπ, and ˆσ—having

the same relative order as π, σ, π, and σ, respectively—satisfying S(πf , σf) = S(π, σ)f ,

S(πf , σf ) = S(π, σ)f , S(ˆπf , ˆσf ) = S(π, σ)f , and S(πf , σf ) = S(ˆπ, ˆσ)f . By the “same relative

order” property, we have

[πf ]st1 = [πf ]st1 = [πf ]st1 = [ˆπf ]st1

and

[σf ]st1 = [σf ]st1 = [σf ]st1 = [ˆσf ]st1 .

Now, by shuffle-compatibility of st1, we have the equality of multisets

{ st1(τ) : τ ∈ S(πf , σf ) } = { st1(τ) : τ ∈ S(ˆπf , ˆσf ) },

which is equivalent to

{ st2(τ) : τ f ∈ S(πf , σf ) } = { st2(τ) : τ f ∈ S(ˆπf , ˆσf ) }

by f -equivalence of st1 and st2, and from S(πf , σf ) = S(π, σ)f and S(ˆπf , ˆσf ) = S(π, σ)f , we

have

{ st2(τ) : τ ∈ S(π, σ) } = { st2(τ) : τ ∈ S(π, σ) }.

Therefore, st2 is shuffle-compatible.

It remains to prove that Ast2 is isomorphic to Ast1 . Observe that∑τ∈S(π,σ)

[τ ]st2 =∑

τ∈S(π,σ)

[τ ]st2 ,

97


since st2 is shuffle-compatible. Define the linear map ϕf : Ast2 → Ast1 by [π]st2 7→ [πf ]st1 .

Then

ϕf ([π]st2 [σ]st2) = ϕf

( ∑τ∈S(π,σ)

[τ ]st2

)=

∑τ∈S(π,σ)

ϕf ([τ ]st2)

=∑

τ∈S(π,σ)

[τ f ]st1

=∑

τ∈S(π,σ)

[τ f ]st1

=∑

τ∈S(π,σ)f

[τ ]st1

=∑

τ∈S(πf ,σf )

[τ ]st1

= [πf ]st1 [σf ]st1

= ϕf ([π]st2)ϕf ([σ]st2),

so ϕf is an isomorphism from Ast2 to Ast1 . �

Lemma 5.7. Reversion, complementation, and reverse-complementation are shuffle-com-

patibility-preserving.

Proof. It is clear that S(πr, σr) = S(π, σ)r, so by taking π = π and σ = σ, the

equalities S(πr, σr) = S(π, σ)r and S(πr, σr) = S(π, σ)r come for free. Thus reversion is

shuffle-compatibility-preserving.

Unlike with reversion, it is not true in general that S(πc, σc) = S(π, σ)c. For disjoint

permutations π = π1π2 · · · πm and σ = σ1σ2 · · ·σn, let P = {π1, . . . , πm, σ1, . . . , σn} be the

set of letters appearing in π and σ, and let ρ : P → P be the map sending the ith smallest

letter of P to the ith largest letter of P for every i. By an abuse of notation, let ρ(π) denote98


the permutation ρ(π1)ρ(π2) · · · ρ(πm) obtained by applying ρ to each letter in π. Then, let

π = ρ(πc) and σ = ρ(σc). For example, let π = 413 and σ = 25. Then P = [5], πc = 143,

and σc = 52, and so π = 523 and σ = 14. Clearly, π has the same relative order as π, and

similarly with σ and σ. It is also easy to see that ρ(π) = πc = πc and ρ(σ) = σc = σc.

To see that S(πc, σc) = S(π, σ)c, first let τ ∈ S(π, σ). Then τ contains both π and σ

as subsequences, and to show that τ c ∈ S(πc, σc), it suffices to show that τ c contains both

πc = ρ(π) and σc = ρ(σ) as subsequences. However, this follows from the fact that, when

taking the complement of τ , the subsequence π appearing in τ is transformed into ρ(π),

and similarly σ turns into ρ(σ). The other inclusion follows by the same reasoning, and the

equality S(πc, σc) = S(π, σ)c follows directly from S(πc, σc) = S(π, σ)c and replacing π and

σ with πc and σc, respectively. Hence complementation is shuffle-compatibility-preserving.

Finally, the equalities S(πr, σr) = S(π, σ)r, S(πc, σc) = S(π, σ)c, and S(πc, σc) = S(π, σ)c

imply S(πrc, σrc) = S(π, σ)rc and S(πrc, σrc) = S(π, σ)rc. Thus reverse-complementation is

shuffle-compatibility-preserving. �

Corollary 5.8. Suppose that st1 and st2 are r-equivalent, c-equivalent, or rc-equivalent

statistics. If st1 is shuffle-compatible with shuffle algebra Ast1, then st2 is also shuffle-

compatible with shuffle algebra Ast2 isomorphic to Ast1.

5.2.3. A note on Hadamard products. The operation of Hadamard product ∗ on

formal power series in t is given by( ∞∑n=0

antn

)∗( ∞∑n=0

bntn

):=

∞∑n=0

anbntn.

Many shuffle algebras that we study can be characterized as subalgebras of various algebras

in which the multiplication is the Hadamard product in a variable t. In the notation for

these algebras, we write t∗ to indicate that multiplication is the Hadamard product in t. For

example, Q[[t∗, q]][x] is the algebra of polynomials in x whose coefficients are formal power

99


series in t and q, where multiplication is ordinary multiplication in the variables x and q but

is the Hadamard product in t.

We note that the Hadamard product is only used in descriptions of shuffle algebras and in

the proof of Lemma 5.17, where tm ∗ tn denotes the Hadamard product of tm and tn. (Here,

tm is the ordinary product of m copies of t and similarly with tn.) All other expressions

should be interpreted as using ordinary multiplication. For instance, any expression with an

exponent such as tk or (1 + yt)k is ordinary multiplication, and (1− tf)−1 (as in Corollary

5.18) denotes∑∞

k=0 tkfk.

5.3. Theory of shuffle-compatibility for descent statistics

5.3.1. Shuffle-compatibility of Des, Pk, and Lpk. Throughout this chapter, fix Q

as our base field, so that QSym is a Q-algebra and Sym is a Q-bialgebra.

Recall from Theorem 2.9 that the fundamental quasisymmetric functions multiply by the

rule

FJFK =∑L

cLJ,KFL, (56)

where cLJ,K is the number of permutations with descent composition L among the shuffles

of a permutation π with descent composition J and a permutation σ (disjoint from π) with

descent composition K. This implies that QSym is isomorphic to the shuffle algebra of the

descent set with the fundamental basis corresponding to the basis of Des-equivalence classes.

Corollary 5.9 (Shuffle-compatibility of the descent set).

(a) The descent set Des is shuffle-compatible.

(b) The linear map on ADes defined by

[π]Des 7→ FComp(π)

is a Q-algebra isomorphism from ADes to QSym.

100


In 1997, John Stembridge [58] introduced a variant of the notion of P -partitions called

“enriched P -partitions”, which is closely related to the combinatorics of peaks. Using enriched

P -partitions, Stembridge defined the peak quasisymmetric functions {Kn,Λ} which are indexed

by peak sets Λ of n-permutations. These peak functions multiply by a rule similar to (56)

but with the role of descent compositions (equivalently, descent sets) replaced with peak sets,

which shows that the peak set Pk is shuffle-compatible with shuffle algebra APk isomorphic to

the span of the peak functions, called the algebra of peaks and denoted Π. Since Pk and Val

are c-equivalent, it follows from Corollary 5.8 that the valley set Val is also shuffle-compatible

and that its shuffle algebra is also isomorphic to the algebra of peaks.

Theorem 5.10 (Shuffle-compatibility of the peak set).

(a) The peak set Pk is shuffle-compatible.

(b) The linear map on APk defined by

[π]Pk 7→ K|π|,Pk(π)

is a Q-algebra isomorphism from APk to Π.

In a similar vein, Kyle Petersen [42] introduced “left enriched P -partitions” which play

an analogous role as enriched P -partitions but for left peaks. It follows from Petersen’s

work that the left peak set Lpk is shuffle-compatible and that the shuffle algebra ALpk is

isomorphic to Petersen’s algebra of left peaks.2 The algebra of left peaks is denoted Π(`) and

are spanned by the left peak quasisymmetric functions {K(`)n,Λ} which are indexed by left peak

sets Λ of n-permutations. We note that the functions {K(`)n,Λ} are not actually quasisymmetric

functions but rather quasisymmetric functions of type B; see [8] for a reference.2Petersen actually calls this algebra the “left algebra of peaks”, but the “algebra of left peaks” seems to us amore natural name.

101


Theorem 5.11 (Shuffle-compatibility of the left peak set).

(a) The left peak set Lpk is shuffle-compatible.

(b) The linear map on ALpk defined by

[π]Lpk 7→ K(`)|π|,Lpk(π)

is a Q-algebra isomorphism from ALpk to Π(`).

Since Lpk and Rpk are r-equivalent, it follows from Corollary 5.8 that the right peak set

Rpk is also shuffle-compatible and that its shuffle algebra is also isomorphic to the algebra of

left peaks.

Although Petersen was the first to explicitly construct the algebra of left peaks, the

shuffle-compatibility of Lpk also follows from the work of Aguiar, Bergeron, and Nyman, who

constructed the coalgebra dual to the algebra of left peaks [2, Proposition 8.3 and Remark

8.7.3]. We will extensively study coalgebras dual to shuffle algebras later in this chapter.

5.3.2. A shuffle-compatibility criterion for descent statistics. Both Stembridge’s

algebra of peaks and Petersen’s algebra of left peaks can be realized as quotients of QSym.

In fact, as a consequence of our next theorem, this is true in general for shuffle algebras of

shuffle-compatible descent statistics.

Let st be a descent statistic. Then not only does st induce a equivalence relation on

permutations, but it also induces a equivalence relation on compositions because permutations

with the same descent composition are necessarily st-equivalent.

Theorem 5.12. A descent statistic st is shuffle-compatible if and only if there exists a

Q-algebra homomorphism φst : QSym→ A, where A is a Q-algebra with basis {uα} indexed

by st-equivalence classes α of compositions, such that φst(FL) = uα whenever L ∈ α. In this

102


case, the linear map on Ast defined by

[π]st 7→ uα,

where Comp(π) ∈ α, is a Q-algebra isomorphism from Ast to A.

Proof. Suppose that st is a shuffle-compatible descent statistic. Let A = Ast be the

shuffle algebra of st, and let uα = [π]st for any π satisfying Comp(π) ∈ α, so that

uβuγ =∑α

cαβ,γuα

where cαβ,γ is the number of permutations with descent composition in α that are obtained as

a shuffle of a permutation π with descent composition in β and a permutation σ (disjoint from

π) with descent composition in γ. Observe that cαβ,γ =∑

L∈α cLJ,K for any choice of J ∈ β and

K ∈ γ, where as before cLJ,K is the number of permutations with descent composition L that

are obtained as a shuffle of a permutation π with descent composition J and a permutation

σ (disjoint from π) with descent composition K.

Define the linear map φst : QSym→ A by φst(FL) = uα for L ∈ α. Then any J ∈ β and

K ∈ γ satisfy

φst(FJFK) = φst

(∑L

cLJ,KFL

)=∑L

cLJ,Kφst(FL)

=∑α

∑L∈α

cLJ,Kuα

=∑α

cαβ,γuα

= uβuγ

= φst(FJ)φst(FK),

103


so φst is a Q-algebra homomorphism, thus completing one direction of the proof. The converse

follows directly from Theorem 5.4. �

It is immediate from Theorem 5.12 that when st is shuffle-compatible, its shuffle algebra

is isomorphic to QSym / ker(φst).

Corollary 5.13. The shuffle algebra of every shuffle-compatible descent statistic is

isomorphic to a quotient algebra of QSym.

5.3.3. A dual shuffle-compatibility criterion for descent statistics. Let st be a

descent statistic. For each st-equivalence class α of compositions, let

rstα :=

∑L∈α

rL.

We call the noncommutative symmetric functions rstα st-ribbons.

The following is the dual version of Theorem 5.12.

Theorem 5.14. A descent statistic st is shuffle-compatible if and only if for every st-

equivalence class α of compositions, there exist constants cαβ,γ for which

∆rstα =

∑β,γ

cαβ,γrstβ ⊗ rst

γ ;

that is, the st-ribbons rstα span a subcoalgebra of Sym. In this case, the cαβ,γ are the structure

constants for Ast.

Proof. By Theorem 2.11, we have a pairing between quasisymmetric functions and

noncommutative symmetric functions for which

〈FL, rJ〉 =

{1, if L = J,

0, otherwise.

Suppose that the st-ribbons rstα span a subcoalgebra of Sym with structure constants

cαβ,γ. Let D be the subcoalgebra spanned by the rstα and let i : D → Sym be the canonical

104


inclusion map, a Q-coalgebra homomorphism. Then i induces a Q-algebra homomorphism

io : QSym→ Do given by

io(FL)(rstα ) =

⟨FL, i(r

stα )⟩

=⟨FL, r

stα

⟩=

{1, if L ∈ α,0, otherwise.

Observe that io(FL) = io(FJ) whenever L and J belong to the same st-equivalence class.

Hence, we can define fα := io(FL) for L ∈ α. Then {fα} is the basis of Do dual to {rstα}, so

fβfγ =∑α

cαβ,γfα.

By Theorem 5.12, st is shuffle-compatible with shuffle algebra isomorphic to Do. We omit the

proof of the reverse implication, as it is similar; we begin with a quotient algebra of QSym

and then show that its basis elements are dual to the st-ribbons rstα . �

While Theorem 5.12 tells us that we can prove the shuffle-compatibility of a descent

statistic by constructing suitable quotients of QSym, Theorem 5.14 tells us that we could,

alternatively, construct suitable subcoalgebras of Sym, and this is what we will do in Section

5.4. Moreover, because it is straightforward to compute coproducts of noncommutative

symmetric functions, Theorem 5.14 is useful for showing that a descent statistic is not

shuffle-compatible and for conjecturing that a statistic is shuffle-compatible, which is not the

case for Theorem 5.12.

As an example, we use Theorem 5.12 to prove the shuffle-compatibility of the partial

descent sets Desi,j. For non-negative integers i and j, let

Desi,j(π) := Des(π) ∩ ({1, 2, . . . , i} ∪ {n− 1, . . . , n− j}),

105


where n = |π|. In other words, Desi,j(π) is the set of descents of π that occur in the first i

or last j positions. For example, if i+ j ≥ |π| − 1 then Desi,j(π) = Des(π), and for |π| ≥ 2,

|Des1,0(π)| = sir(π) and |Des0,1(π)| = sfr(π), with sir and sfr as defined in Section 2.1.3

Theorem 5.15. The partial descent sets Desi,j for all i, j ≥ 0 are shuffle-compatible.

Proof. Fix i, j ≥ 0. For a fixed n ≥ 0, the set {rDesi,jα } over Desi,j-equivalence classes α

corresponding to n-permutations is given by

{rDesi,jα } =

{{rJhm1 rK}J�i+1,K�j+1,m=n−i−j−2, if n ≥ i+ j + 2,

{rL}L�n, otherwise.

Let V denote the subspace of Sym spanned by the Desi,j-ribbons rDesi,jα over all n ≥ 0. By

Theorem 5.14, it suffices to show that V is a subcoalgebra of Sym, i.e., ∆V ⊆ V ⊗ V .

First, consider the case rDesi,jα = rL where L � n for some n < i+j+2. Then ∆(rL) can be

written as a linear combination of tensor products of the form rL′ ⊗ rL′′ where |L′| < i+ j + 2

and |L′′| < i+ j + 2. Hence, ∆(rL) ∈ V ⊗ V .

Next, consider the case rDesi,jα = rJh

m1 rK where J � i+1, K � j+1, andm = n−i−j−2 ≥

0. We know that the ribbons are closed under comultiplication, so we can write ∆(rJ) as

a linear combination of tensors of the form rJ ′ ⊗ rJ ′′ and ∆(rK) as a linear combination of

tensors of the form rK′ ⊗ rK′′ . Moreover, ∆(h1) = 1⊗ h1 + h1 ⊗ 1. Thus,

∆(rJhm1 rK) = ∆(rJ)∆(h1)m∆(rK)

=( ∑J ′,J ′′

rJ ′ ⊗ rJ ′′)

(1⊗ h1 + h1 ⊗ 1)m( ∑K′,K′′

rK′ ⊗ rK′′)

is a linear combination of tensors of the form rJ ′hk1rK′ ⊗ rJ ′′h

m−k1 rK′′ . To show that

∆(rJhm1 rK) ∈ V ⊗ V , we show that each of these terms belong to V ⊗ V .

3In fact, it is easy to see that Des1,0 is equivalent to sir and that Des0,1 is equivalent to sfr. Thus, by Theorem5.3, Theorem 5.15 implies that sir and sfr are shuffle-compatible as well.

106


Fix a term rJ ′hk1rK′⊗rJ ′′h

m−k1 rK′′ . Let n1 = |J ′|+k+ |K ′| and n2 = |J ′|+m−k+ |K ′|. If

both n1 < i+j+2 and n2 < i+j+2, then we write rJ ′hk1rK′ as a linear combination of ribbons

rM ′ with M ′ � n1 and rJ ′′hm−k1 rK′′ as a linear combination of ribbons rM ′′ with M ′′ � n2.

Thus, when n1 < i+ j + 2 and n2 < i+ j + 2, we have rJ ′hk1rK′ ⊗ rJ ′′h

m−k1 rK′′ ∈ V ⊗ V .

Suppose that n1 ≥ i + j + 2. Then we write rJ ′hk1rK′ = (rJ ′h

u1)hk−u−v1 (hv1rK′) where

|J ′| + u = i + 1 and v + |K ′| = j + 1. Both rJ ′hu1 and hv1rK′ can be written as a linear

combination of ribbons, so rJ ′hk1rK′ is a linear combination of Desi,j-ribbons. If n2 ≥ i+ j+2,

then we can write rJ ′′hm−k1 rK′′ as a linear combination of Desi,j-ribbons in the same way.

Therefore, rJ ′hk1rK′ ⊗ rJ ′′hm−k1 rK′′ ∈ V ⊗ V when n1 ≥ i+ j + 2 or n2 ≥ i+ j + 2, and we

are done. �

5.3.4. Monoidlike noncommutative symmetric functions. Although Theorem 5.14

does not give us a way to describe the dual algebra Ast, we can describe Ast explicitly using

a result that we will present in Section 5.3.5. In doing so, we will often work with noncom-

mutative symmetric functions with coefficients in either the ring Q[x, y] of polynomials in

x and y with rational coefficients or the ring Q[[t∗]][x, y] of polynomials in x and y with

coefficients in the ring of formal power series in t in which multiplication is the Hadamard

product in t but ordinary multiplication in x and y. We will also need to use formal sums of

noncommutative symmetric functions of unbounded degree with these coefficient rings, for

example, h(x) =∑∞

n=0 hnxn. We will use the notation Symxy for the algebra of noncommu-

tative symmetric functions of unbounded degree with coefficients in Q[x, y] and Symtxy for

noncommutative symmetric functions with coefficients in Q[[t∗]][x, y]. The comultiplication

∆ of Sym extends naturally to Symxy and Symtxy (but note that now tensor products are

over the coefficient ring), so Symxy and Symtxy are bialgebras.

We will need to consider “monoidlike elements” of these bialgebras. We call an element

f of a bialgebra monoidlike if ∆f = f ⊗ f . It is straightforward to show that the product

107


of two monoidlike elements is monoidlike and that the inverse of a monoidlike element, if it

exists, is monoidlike.4

Lemma 5.16. h(x), e(x), and e(xy) are monoidlike in Symxy.

Proof. We have

∆h(x) =∞∑n=0

∆hnxn

=∞∑n=0

∑i+j=n

(hi ⊗ hj)xn

=∞∑n=0

∑i+j=n

hixi ⊗ hjx

j

=∞∑

i,j=0

hixi ⊗ hjx

j

=( ∞∑i=0

hixi)⊗( ∞∑j=0

hjxj),

so h(x) is monoidlike. Since e(x) = h(−x)−1, this implies that e(x) and e(xy) are monoidlike.

�

Lemma 5.17. Let f =∑∞

n=0 antn be an element of Symtxy where each an is an element

of Symxy. Then f is monoidlike in Symtxy if and only if each an is monoidlike in Symxy.

Proof. We have

f ⊗ f =∞∑

m,n=0

amtm ⊗ antn

=∞∑

m,n=0

(am ⊗ an)(tm ∗ tn)

4A monoidlike element f of a bialgebra is called grouplike if ε(f) is the identity element of the coefficientring, where ε is the counit. In our bialgebras, the counit is the coefficient of h0, the identity element of Qor Q[x, y] is 1, and the identity element of Q[[t∗]][x, y] is (1− t)−1 =

∑∞k=0 t

k. Nearly all of our monoidlikeelements are actually grouplike, but exceptions occur in Corollary 5.18.

108


=∞∑n=0

(an ⊗ an)tn

and

∆f =∞∑n=0

∆antn.

Thus ∆f = f ⊗ f if and only if ∆an = an ⊗ an for each n. �

The next result follows immediately from Lemma 5.17.

Corollary 5.18. Suppose that f is monoidlike in Symxy. Then (1− tf)−1, (1− t2f)−1,

and 1 + tf are monoidlike in Symtxy.

5.3.5. Monoidlike elements and shuffle-compatibility. For an st-equivalence class

α of compositions, we let |α| be equal to |L| for any composition L ∈ α.

Theorem 5.19. Let st be a descent statistic and let uα ∈ Q[[t∗]][x, y] be linearly inde-

pendent elements (over Q) indexed by st-equivalence classes α of compositions. Suppose

that f =∑

α uαrstα is monoidlike in Symtxy and that there exist constants cαβ,γ such that

uβuγ =∑

α cαβ,γuα for all st-equivalence classes β and γ, where cαβ,γ = 0 unless |α| = |β|+ |γ|.

Then st is shuffle-compatible and the linear map defined by

[π]st 7→ uα,

where Comp(π) ∈ α, is a Q-algebra isomorphism from Ast to the subalgebra of Q[[t∗]][x, y]

spanned by the uα.

Proof. Since f is monoidlike, we have that∑α

uα∆rstα = ∆f =

(∑β

uβrstβ

)⊗(∑

γ

uγrstγ

)=∑β,γ

uβuγrstβ ⊗ rst

γ

109


=∑α

uα∑β,γ


γ .

Extracting the linear combinations of elements of Symi ⊗ Symj, where i + j = n, we

obtain ∑|α|=n

uα∆rstα =

∑|α|=n

uα∑β,γ


γ .

Since these are finite sums, linear independence of the uα implies

∆rstα =

∑β,γ


γ

and it follows from Theorem 5.14 that st is shuffle-compatible and that the cαβ,γ are the

structure constants for Ast. Since

uβuγ =∑α

cαβ,γuα

for all st-equivalence classes β and γ, the map [π]st 7→ uα is an algebra homomorphism

from Ast to the subalgebra of Q[[t∗]][x, y] spanned by the uα, and since the uα are linearly

independent, this map is an isomorphism. �

We note that Theorem 5.19 can be generalized to a statement about monoidlike elements

of more general graded bialgebras; we stated it only in the special case that we will use.

Unfortunately, in our applications, it is difficult to show directly that the desired uα are

closed under multiplication. The following variant of Theorem 5.19 uses a change of basis

argument to deal with this problem.

Theorem 5.20. Let st be a descent statistic and let uα ∈ Q[[t∗]][x, y] be linearly indepen-

dent elements (over Q) indexed by st-equivalence classes α of compositions. Suppose that

f =∑

α uαrstα is monoidlike in Symtxy, where uα is x|α| times an element of Q[[t∗]][y]. Let

sn,p,q be the coefficient of xnyptq in∑

α uαrstα and suppose that rst

α ∈ SpanQ{sn,p,q} for each α.

110


Then st is shuffle-compatible and the linear map defined by

[π]st 7→ uα,

where Comp(π) ∈ α, is a Q-algebra isomorphism from Ast to the subalgebra of Q[[t∗]][x, y]

spanned by the uα.

Proof. Equating coefficients of xn in

f =∑α

uαrstα =

∑n,p,q

xnyptqsn,p,q

gives ∑|α|=n

uαrstα = xn

∑p,q

yptqsn,p,q.

Since the sum on the left is finite, this shows that sn,p,q ∈ SpanQ{rstα}, so SpanQ{rst

α} =

SpanQ{sn,p,q}.

Let fq be the coefficient of tq in f . Then since f is monoidlike, fq is monoidlike by Lemma

5.17, so ∑n,p

xnyp∆sn,p,q = ∆fq = fq ⊗ fq

=(∑n1,p1

xn1yp1sn1,p1,q

)⊗(∑n2,p2

xn2yp2sn2,p2,q

)=

∑n1,p1,n2,p2

xn1+n2yp1+p2sn1,p1,q ⊗ sn2,p2,q

Equating coefficients of xnyp shows that SpanQ{sn,p,q} is a subcoalgebra of Sym and thus so

is SpanQ{rstα}. As a result, there exist constants cαβ,γ such that

∆rstα =

∑β,γ


γ ,

so it follows from Theorem 5.14 that st is shuffle-compatible and that the cαβ,γ are the structure

constants for Ast.

111


Moreover, since∑

α uαrstα is monoidlike, we have∑

β,γ

∑α

uαcαβ,γr

stβ ⊗ rst

γ =∑α

uα∑β,γ


γ

=∑α

uα∆rstα

= ∆(∑

α

uαrstα

)=(∑

β

uβrstβ

)⊗(∑

γ

uγrstγ

)=∑β,γ

uβuγrstβ ⊗ rst

γ .

Using the linear independence of the rstβ ⊗ rst

γ and the fact that for each i and j, rstβ ⊗ rst

γ ∈

Symi⊗Symj for only finitely many β and γ, we may equate coefficients of rstβ ⊗ rst

γ to obtain

uβuγ =∑

α cαβ,γuα. Thus the map [π]st 7→ uα is an algebra homomorphism from Ast to the

subalgebra of Q[[t∗]][x, y] spanned by the uα, and since the uα are linearly independent, this

map is an isomorphism. �

5.4. Explicit descriptions of shuffle algebras

5.4.1. Shuffle-compatibility of pk and (pk, des). We will use Theorem 5.20 to char-

acterize (i.e., give explicit descriptions of) the shuffle algebras of the permutation statistics

pk, (pk, des), lpk, (lpk, des), udr, (udr, des), des, and (des,maj), thus showing that they are

all shuffle-compatible. All computations are done in the algebra Symtxy of noncommutative

symmetric functions with coefficients in Q[[t∗]][x, y].

We begin with the peak number pk and the pair (pk, des), first stating the result for

(pk, des) and then deriving from it the result for pk using Theorem 5.4.

Theorem 5.21 (Shuffle-compatibility of (pk, des)).

(a) The pair (pk, des) is shuffle-compatible.

112


(b) The linear map on A(pk,des) defined by

[π](pk,des) 7→tpk(π)+1(y + t)des(π)−pk(π)(1 + yt)|π|−pk(π)−des(π)−1(1 + y)2 pk(π)+1

(1− t)|π|+1x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

is a Q-algebra isomorphism from A(pk,des) to the span of{1

1− t

}⋃{tj+1(y + t)k−j(1 + yt)n−j−k−1(1 + y)2j+1

(1− t)n+1xn}n≥1,0≤j≤b(n−1)/2c,j≤k≤n−j−1

,

a subalgebra of Q[[t∗]][x, y].

(c) The (pk, des) shuffle algebra A(pk,des) is isomorphic to the span of

{1} ∪ {pn−j(1 + y)n(1− y)n−2kxn}n≥1, 0≤j≤n−1, 0≤k≤bj/2c,

a subalgebra of Q[p, x, y].

(d) For n ≥ 1, the nth homogeneous component of A(pk,des) has dimension b(n+ 1)2/4c.

We prove here parts (a), (b), and (d). We omit the proof of part (c), but refer the

interested reader to [22, Section 6.1].

Proof. Let us write r(pk,des)n,j,k for the (pk, des)-ribbon r

(pk,des)α where α is the (pk, des)-

equivalence class of compositions corresponding to n-permutations with j− 1 peaks and k− 1

descents. By Lemma 2.5 and Proposition 1.5, we have

(1− te(xy)h(x))−1

=1

1− t+∞∑n=1

b(n−1)/2c∑j=0

n−j−1∑k=j

tj+1(y + t)k−j(1 + yt)n−j−k−1(1 + y)2j+1

(1− t)n+1xnr

(pk,des)n,j+1,k+1

=1

1− t+∞∑n=1

b(n+1)/2c∑j=1

n−j+1∑k=j

tj(y + t)k−j(1 + yt)n−j−k+1(1 + y)2j−1

(1− t)n+1xnr

(pk,des)n,j,k ,

113


and this is monoidlike by Lemma 5.16 and Corollary 5.18. Now define sn,p,q by∞∑

n,p,q=0

xnyptqsn,p,q = (1− te(xy)h(x))−1.

For fixed n ≥ 1, we have

∞∑p,q=0

yptqsn,p,q =

b(n+1)/2c∑j=1

n−j+1∑k=j

tj(y + t)k−j(1 + yt)n−j−k+1(1 + y)2j−1

(1− t)n+1r

(pk,des)n,j,k .

This identity can be inverted to obtainb(n+1)/2c∑

j=1

n−j+1∑k=j

yjtkr(pk,des)n,j,k = (1 + u)

(1− v1 + uv

)n+1 ∞∑p,q=0

upvqsn,p,q,

where

u =1 + t2 − 2yt− (1− t)

√(1 + t)2 − 4yt

2(1− y)t

and

v =(1 + t)2 − 2yt− (1 + t)

√(1 + t)2 − 4yt

2yt,

in the formal power series ring Q[[t, y]]. It is easily checked that u and v are both formal

power series divisible by t, so (1− v)/(1 + uv) is a well-defined formal power series in t and y.

Equating coefficients of yptq shows that each r(pk,des)n,j,k is a linear combination of the sn,p,q.

(Since u and v are divisible by t, only finitely many terms on the right will contribute a term

in tq.) Parts (a) and (b) then follow from Theorem 5.20.

By Proposition 1.5, we know that for n ≥ 1, the number of (pk, des)-equivalence classes

for n-permutations isb(n−1)/2c∑

j=0

((n− j − 1)− j + 1) =

b(n−1)/2c∑j=0

(n− 2j),

which is easily shown to be equal to b(n+ 1)2/4c. This proves (d). �

Note that (pk, des) and (val, des) are rc-equivalent statistics, and that (val, des) and

(epk, des) are equivalent statistics. Thus, by Corollary 5.8 and Theorem 5.3, we know that

114


(val, des) and (epk, des) are also shuffle-compatible and have shuffle algebras isomorphic to

A(pk,des).

Theorem 5.22 (Shuffle-compatibility of the peak number).

(a) The peak number pk is shuffle-compatible.

(b) The linear map on Apk defined by

[π]pk 7→

22 pk(π)+1tpk(π)+1(1 + t)|π|−2 pk(π)−1

(1− t)|π|+1x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

is a Q-algebra isomorphism from Apk to the span of{1

1− t

}⋃{22j+1tj+1(1 + t)n−2j−1

(1− t)n+1xn}n≥1, 0≤j≤bn−1

2 c,

a subalgebra of Q[[t∗]][x].

(c) The pk shuffle algebra Apk is isomorphic to the span of

{1} ∪ {pjxn}n≥1, 1≤j≤n, j≡n (mod 2),

a subalgebra of Q[p, x].

(d) For n ≥ 1, the nth homogeneous component of Apk has dimension b(n+ 1)/2c.

Again, we prove here only parts (a), (b), and (d). The proof for part (c) can be found in

[22, Section 6.1].

Proof. Let φ be the homomorphism from Q[[t∗]][x, y] to Q[[t∗]][x] obtained by setting y

to 1. It is easy to check that φ takes the image of [π](pk,des) as described in Theorem 5.21 (b)

to the image of [π]pk as given in (b). Then (a) and (b) follow from Theorem 5.4.

Part (d) follows from Proposition 1.7. �

By Corollary 5.8, Lemma 1.3 (e), and Theorem 5.3, the valley number val and exterior

peak number epk are also shuffle-compatible and have shuffle algebras isomorphic to Apk.

115


5.4.2. Shuffle-compatibility of lpk and (lpk, des). We now prove that the left peak

number lpk and the pair (lpk, des) are shuffle-compatible. We first prove the result for

(lpk, des) and then derive the shuffle-compatibility of lpk from that of (lpk, des).

Theorem 5.23 (Shuffle-compatibility of (lpk, des)).

(a) The pair (lpk, des) is shuffle-compatible.

(b) The linear map on A(lpk,des) defined by

[π](lpk,des) 7→tlpk(π)(y + t)des(π)−lpk(π)(1 + yt)|π|−lpk(π)−des(π)(1 + y)2 lpk(π)

(1− t)|π|+1x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

is a Q-algebra isomorphism from A(lpk,des) to the span of{1

1− t

}⋃{ (1 + yt)n

(1− t)n+1xn}n≥1

⋃{tj(y + t)k−j(1 + yt)n−j−k(1 + y)2j

(1− t)n+1xn}n≥2,1≤j≤bn/2c,j≤k≤n−j

,

a subalgebra of Q[[t∗]][x, y].

(c) The nth homogeneous component of A(lpk,des) has dimension bn2/4c+ 1.

Proof. Let r(lpk,des)n,j,k denote r

(lpk,des)α where α is the (lpk, des)-equivalence class of com-

positions corresponding to n-permutations with j left peaks and k descents. Define sn,p,q

by∞∑

n,p,q=0

xnyptqsn,p,q = h(x)(1− te(xy)h(x))−1.

Then the proofs for parts (a) and (b) follow in the same manner as for Theorem 5.21, using

Proposition 1.6, Lemma 2.6, and Corollary 5.18 along the way.

By Proposition 1.6, the number of (lpk, des)-equivalence classes for n-permutations is

1 +

bn/2c∑j=1

((n− j)− j + 1) = 1 +

bn/2c∑j=1

(n− 2j + 1),

116


which is easily shown to be equal to bn2/4c+ 1. This proves (c). �

Although (lpk, des) and (rpk, des) are not equivalent, r-equivalent, c-equivalent, or rc-

equivalent, this argument does show that (rpk, des) is shuffle-compatible and has shuffle

algebra isomorphic to that of (lpk, des) because (lpk, des) is r-equivalent to (rpk, asc)—where

asc is the number of ascents—and (rpk, asc) is equivalent to (rpk, des).

Theorem 5.24 (Shuffle-compatibility of the left peak number).

(a) The left peak number lpk is shuffle-compatible.

(b) The linear map on Alpk defined by

[π]lpk 7→

22 lpk(π)tlpk(π)(1 + t)|π|−2 lpk(π)

(1− t)|π|+1x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

is a Q-algebra isomorphism from Alpk to the span of{1

1− t

}⋃{22jtj(1 + t)n−2j

(1− t)n+1xn}n≥1, 0≤j≤bn/2c

,


(c) The nth homogeneous component of Alpk has dimension bn/2c+ 1.

The proof uses Theorem 5.4 and follows in the same way as the proof of Theorem 5.22.

By Lemma 1.3 (d) and Theorem 5.8, the right peak number rpk and the number of long

runs lr are also shuffle-compatible and have shuffle algebras isomorphic to that of lpk.

5.4.3. Shuffle-compatibility of udr and (udr, des). Next, we prove the analogous

results for the number of up-down runs udr and the pair (udr, des). As before, we first prove

the result for (udr, des) and then use Theorem 5.4 to derive the result for udr.

Theorem 5.25 (Shuffle-compatibility of (udr, des)).

(a) The pair (udr, des) is shuffle-compatible.

117


(b) The linear map on A(udr,des) defined by

[π](udr,des) 7→

Nπ

(1− t)(1− t2)|π|x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

where

Nπ = tudr(π)(1 + y)udr(π)−1(1 + yt2)|π|−des(π)−dudr(π)/2e(y + t2)des(π)−budr(π)/2c

× (1 + yt)dudr(π)/2e−budr(π)/2c(y + t)1−dudr(π)/2e+budr(π)/2c,

is a Q-algebra isomorphism from A(udr,des) to the span of{1

1− t

}⋃{t(1 + yt)(1 + yt2)n−1

(1− t)(1− t2)nxn}n≥1⋃{

tj(1 + y)j−1(1 + yt2)n−k−dj/2e(y + t2)k−bj/2cSj(1− t)(1− t2)n

xn}n≥1,2≤j≤n,bj/2c≤k≤n−dj/2e

,

where Sj is 1 + yt if j is odd and is y + t if j is even, a subalgebra of Q[[t∗]][x, y].

(c) The nth homogeneous component of A(udr,des) has dimension(n2

)+ 1.

Proof. By Lemma 2.7, together with Lemma 1.4 (b) and (c), we have

(1− t2h(x)e(xy))−1(1 + th(x)) =1

1− t+∞∑n=1

∑L�n

NL

(1− t)(1− t2)nxnrL (57)

where

NL = tudr(L)(1 + y)udr(L)−1(1 + yt2)n−des(L)−dudr(L)/2e(y + t2)des(L)−budr(L)/2c

× (1 + yt)dudr(L)/2e−budr(L)/2c(y + t)1−dudr(L)/2e+budr(L)/2c.

Note that dudr(L)/2e − budr(L)/2c is 1 if udr(L) is odd and is 0 if udr(L) is even. The

left-hand side of (57) is monoidlike by Lemma 5.16 and Corollary 5.18.

Let r(udr,des)n,j,k denote r

(udr,des)α where α is the (udr, des)-equivalence class of compositions

corresponding to n-permutations with j up-down runs and k descents. Then by (57) and

118


Proposition 1.7, we have

(1− t2h(x)e(xy))−1(1 + th(x)) =1

1− t+∞∑n=1

(t(1 + yt)(1 + yt2)n−1

(1− t)(1− t2)nxnr

(udr,des)n,1,0

+∑

2≤j≤nbj/2c≤k≤n−dj/2e

tj(1 + y)j−1(1 + yt2)n−k−dj/2e(y + t2)k−bj/2cSj(1− t)(1− t2)n

xnr(udr,des)n,j,k

)(58)

with Sj as in the statement of the theorem. Define sn,p,q by∞∑

n,p,q=0

xnyptqsn,p,q = (1− t2h(x)e(xy))−1(1 + th(x)). (59)

To prove (a) and (b), as in Theorems 5.21 and 5.23, it is sufficient to show that each r(udr,des)n,j,k

is in the span of the sn,p,q. Because of the floor and ceiling functions in (58), we are not able

to use the generating function inversion method that we used in the proofs of Theorems 5.21

and 5.23, so we take a different approach.

Expanding the right side of (58) and comparing with (59) shows that, for fixed n, each

sn,p,q is a linear combination (with integer coefficients) of the r(udr,des)n,j,k . We will show that

these relations can be inverted to express each r(udr,des)n,j,k as a linear combination of the sn,p,q.

We totally order N × N colexicographically, so (p1, q1) ≤ (p2, q2) if and only if q1 < q2

or q1 = q2 and p1 ≤ p2. We shall show that for each j and k, there exist p and q such that

r(udr,des)n,j,k appears with coefficient 1 in sn,p,q and if r(udr,des)

n,j′,k′ appears in sn,p,q then (k′, j′) ≤ (k, j).

This will imply, by induction, that r(udr,des)n,j,k is in SpanQ{sn,p,q}.

With this total order, the monomial yptq with minimal (p, q) that appears in the coefficient

of xnr(udr,des)n,j,k on the right side of (58) is easily seen to be ykj tj (with coefficient 1), where kj

is k−bj/2c+ 1 if j is even and is k−bj/2c if j is odd. In other words, sn,p,q does not contain

any r(udr,des)n,j,k for which (p, q) < (kj, j). Replacing p and q with kj and j, and replacing k and

j with k′ and j′, we have that

sn,kj ,j = r(udr,des)n,j,k +

∑j′, k′

cj′, k′r(udr,des)n,j′,k′

119


where cj′, k′ = 0 unless (k′j′ , j′) < (kj, j). It is easy to see that (k′j′ , j

′) < (kj, j) implies

(k′, j′) < (k, j), so we have

sn,kj ,j = r(udr,des)n,j,k +

∑(k′, j′)<(k,j)

cj′, k′r(udr,des)n,j′,k′

and this completes the proof of (b).

By Proposition 1.7, the number of (udr, des)-equivalence classes for n-permutations is

1 +n∑j=2

(n− bj/2c − dj/2e+ 1) = 1 +n∑j=2

(n− j + 1) = 1 +

(n

2

).

This proves part (c). �

We know from Lemma 1.4 that udr and (lpk, val) are equivalent statistics, from Lemma

1.3 (d) that val is equivalent to epk, and from Proposition 5.5 that (lpk, val) is rc-equivalent

to (lpk, pk). It follows that (udr, des) is equivalent to (lpk, val, des) and (lpk, epk, des), and

is rc-equivalent to (lpk, pk, des). Thus, by Theorem 5.3 and Corollary 5.8, the statistics

(lpk, val, des), (lpk, epk, des), and (lpk, pk, des) are all shuffle-compatible and have shuffle

algebras isomorphic to A(udr,des).

Theorem 5.26 (Shuffle-compatibility of the number of up-down runs).

(a) The number of up-down runs udr is shuffle-compatible.

(b) The linear map on Audr defined by

[π]udr 7→

2udr(π)−1tudr(π)(1 + t2)|π|−udr(π)

(1− t)2(1− t2)|π|−1x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

is a Q-algebra isomorphism from Audr to the span of{1

1− t

}⋃{ 2j−1tj(1 + t2)n−j

(1− t)2(1− t2)n−1xn}n≥1, 1≤j≤n

,


(c) For n ≥ 1, the nth homogeneous component of Audr has dimension n.

120


The proof follows in the same way as the proof of Theorem 5.22.

Since udr and (lpk, val) are equivalent statistics, (lpk, val) is shuffle-compatible and

A(lpk,val) is isomorphic to Audr. Furthermore, since (lpk, val) is rc-equivalent to (lpk, pk), we

have also proven the shuffle-compatibility of (lpk, pk) and characterized the shuffle algebra

A(lpk,pk). Similar reasoning implies that (lpk, epk), (rpk, val), (rpk, pk), (rpk, epk), (lr, val),

(lr, pk), and (lr, epk) are shuffle-compatible and that their shuffle algebras are all isomorphic

to Audr.

5.4.4. Shuffle-compatibility of des and (des,maj). We know that the statistics des

and (des,maj) are shuffle-compatible due to Stanley’s shuffling theorem. To conclude this

section, we characterize the shuffle algebras of these two statistics.

We denote the set of non-negative integers by N.

Theorem 5.27 (Shuffle-compatibility of (des,maj)).

(a) The ordered pair (des,maj) is shuffle-compatible.

(b) The linear map on A(des,maj) defined by

[π](des,maj) 7→ qmaj(π)

(p− des(π) + |π| − 1

|π|

)q

x|π|

is a Q-algebra isomorphism from A(des,maj) to the span of

{1}⋃{

qk(p− j + n− 1

n

)q

xn

}n≥1, 0≤j≤n−1, (j+1

2 )≤k≤nj−(j+12 )

,

a subalgebra of Q[q, x]N, the algebra of functions N → Q[q, x] in the non-negative

integer variable p.

(c) The linear map on A(des,maj) defined by

[π](des,maj) 7→

qmaj(π)tdes(π)+1

(1− t)(1− qt) · · · (1− q|π|t)x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

121


is a Q-algebra isomorphism from A(des,maj) to the span of{1

1− t

}⋃{ qktj+1

(1− t)(1− qt) · · · (1− qnt)xn}n≥1, 0≤j≤n−1 ,(j+1

2 )≤k≤nj−(j+12 ),

a subalgebra of Q[[t∗, q]][x].

(d) For n ≥ 1, the nth homogeneous component of A(des,maj) has dimension(n3

)+ n.

Our proof of this theorem in [22] uses quasisymmetric functions via Theorem 5.12,

although it is also possible to prove the result using noncommutative symmetric functions

via Theorem 5.20 together with [23, Equation 25]. We omit the proof here.

Theorem 5.28 (Shuffle-compatibility of the descent number).

(a) The descent number des is shuffle-compatible.

(b) The linear map on Ades defined by

[π]des 7→(p− des(π) + |π| − 1

|π|

)x|π|

is a Q-algebra isomorphism from Ades to the span of

{1}⋃{(p− j + n− 1

n

)xn}n≥1, 0≤j≤n−1

,


(c) Ades is isomorphic to the span of

{1} ∪ {pjxn}n≥1, 1≤j≤n,


(d) The linear map on Ades defined by

[π]des 7→

tdes(π)+1

(1− t)|π|+1x|π|, if |π| ≥ 1,

1/(1− t), if |π| = 0,

122


is a Q-algebra isomorphism from Ades to the span of{1

1− t

}⋃{ tj+1

(1− t)n+1xn}n≥1, 0≤j≤n−1

,


(e) For n ≥ 1, the nth homogeneous component of Ades has dimension n.

Proof. Applying Theorem 5.4 to Theorem 5.27 with the homomorphism that takes q

to 1, together with the observation that polynomial functions in characteristic zero may be

identified with polynomials, yields (a), (b), and (d). Parts (c) and (e) follow easily from

(b). �

5.5. Non-shuffle-compatible permutation statistics

Although many well-known descent statistics have been shown to be shuffle-compatible,

there are many descent statistics that are not shuffle-compatible. Here we list some of them.

Theorem 5.29. The statistics Pk∪Val, (pk, val), (pk, val, des), (Pk, des), (Pk, val),

(Pk, val, des), (Pk,Val), (Lpk, des), (Lpk, val, des), and (Epk, des) are not shuffle-compatible.

Recall that a birun of a permutation is a maximal monotone consecutive subsequence, and

that br(π) is the number of biruns of π. The number of biruns is not shuffle-compatible, and

the only joint statistics involving br that we have found that seem to be shuffle-compatible

are (Lpk, br) and (Epk, br); however, these are easily shown to be equivalent to Epk, which

is shuffle-compatible (see the discussion following Conjecture 5.32).

Theorem 5.30. The statistics br, (br, des), (br,maj), (br, des,maj), (br, pk), (br, pk, des),

(br, lpk), (br, lpk, des), and (Pk, br) are not shuffle-compatible.

Although (des,maj) is shuffle-compatible, we have not found any other shuffle-compatible

joint statistics involving the major index.123


Theorem 5.31. The statistics (pk,maj), (lpk,maj), (Pk,maj), (Lpk,maj), (udr,maj),

(pk, des,maj), (lpk, des,maj), (udr, des,maj), and (lir,maj) are not shuffle-compatible.

In addition to the descent statistics examined in this chapter, we have studied two

additional families of descent statistics, one based on the notion of double ascents, and one

based on the notion of alternating descents. In addition to those that were defined in Section

1.2, these also include the double ascent set, right double ascent set, exterior double ascent

set, and alternating major index, which are all defined in the obvious way. Aside from the

alternating descent set—which is equivalent to the descent set—none of these statistics are

shuffle-compatible. Among joint statistics that involve one or more of these statistics, we

have not found any that seem to be shuffle-compatible (other than a few that are equivalent

to statistics that we know to be shuffle-compatible).

Lastly, among permutation statistics that are not descent statistics, we have not found

any that seem to be shuffle-compatible.

5.6. Open problems and conjectures

We now state a couple permutation statistics that we conjecture to be shuffle-compatible

based on empirical evidence, and present a few more general open problems and conjectures

on the topic of shuffle-compatibility.

Conjecture 5.32. The statistics (udr, pk) and (udr, pk, des) are shuffle-compatible.

In a previous version of [22], we conjectured that the exterior peak set Epk is shuffle-

compatible, along with the tuples (Pk, val, des), (Pk, udr), (Lpk, val), and (Lpk, val, des). All

of these have been addressed by Darij Grinberg. Specifically, Grinberg proved that Epk is

shuffle-compatible using a P -partition argument [27], noted that (Pk, udr) and (Lpk, val)

are both equivalent to Epk, and found counterexamples showing that (Pk, val, des) and

(Lpk, val, des) are not shuffle-compatible [29].124


Prior to this, Grinberg had shown that QSym is a “dendriform algebra” [28], an algebra

whose multiplication can be split into a “left multiplication” and a “right multiplication”

satisfying certain nice axioms. Together with the shuffle-compatibility of Epk, Grinberg

proved that AEpk is a dendriform quotient of QSym. More generally, he proved that a descent

statistic is a dendriform quotient of QSym if and only if it is both “left-shuffle-compatible”

and “right-shuffle-compatible”, which are combinatorial conditions that, together, refine the

notion of shuffle-compatibility. Other descent statistics that Grinberg has shown to be both

left- and right-shuffle-compatible include the descent number des, the pair (des,maj), and

the left peak set Lpk. On the other hand, the major index maj, the peak set Pk, and the

right peak set Rpk are neither left- nor right-shuffle-compatible.

From Theorem 5.29, we know that a pair of two shuffle-compatible statistics need not be

shuffle-compatible. Hence, we pose the following question.

Question 5.33. Suppose that st1 and st2 are shuffle-compatible statistics. Are there

simple conditions that imply that the pair (st1, st2) is shuffle-compatible?

Similarly, if a pair is shuffle-compatible, then that does not imply that the individual

statistics in the pair are both shuffle-compatible.

Question 5.34. Suppose that the pair (st1, st2) is shuffle-compatible. Are there simple

conditions that imply that st1 and st2 are both shuffle-compatible?

Finally, we present the following conjecture.

Conjecture 5.35. Every shuffle-compatible permutation statistic is a descent statistic.

5.7. Two remarks: the Malvenuto–Reutenauer algebra and the descent algebra

We note that some permutation statistics, such as the inversion number inv, satisfy a

weak form of shuffle-compatibility: for disjoint permutations π and σ, if every letter of π is125

CHAPTER . SHUFFLE-COMPATIBLE PERMUTATION STATISTICS

less than every letter of σ, then the distribution of st over S(π, σ) depends only on st(π),

st(σ), |π|, and |σ|. Permutation statistics with this property are associated with quotients of

the Malvenuto–Reutenauer algebra (also called the algebra of free quasisymmetric functions).

Some of these statistics have been studied by Vong [59], but we do not consider this weak

form of shuffle-compatibility here.

Also, there is another class of algebras that are related to permutations and their descent

sets, based on ordinary multiplication of permutations rather than shuffles. If st is a function

defined on the nth symmetric group Sn, we may consider the elements

Kα =∑π∈Sn

st(π)=α

π

in the group algebra of Sn, where α ranges over the image of st. Louis Solomon [49] proved

that if st is the descent set, then the Kα span a subalgebra of the group algebra of Sn, called

the descent algebra of Sn. Several other descent statistics give subalgebras of the descent

algebra, including the descent number [36]; the peak set [40, 46]; the left peak set, peak

number, and left peak number [2, 41, 42]; and the number of biruns and up-down runs

[11, 32]. These descent statistics have the property that given values α and β of st, and

τ ∈ Sn, the number of pairs (π, σ) of permutations in Sn with st(π) = α, st(σ) = β, and

πσ = τ depends only on st(τ). In other words, these statistics are “compatible” under the

ordinary product, and our work is an analogue of Solomon’s descent theory for statistics

compatible under the shuffle product.

Although there is a significant overlap between shuffle-compatible permutation statistics

and statistics corresponding to subalgebras of the descent algebra, neither class is contained

in the other, as the number of biruns is not shuffle-compatible and the pair (pk, des) does not

give a subalgebra of the descent algebra. The descent algebra and its subalgebras may also

126

CHAPTER . SHUFFLE-COMPATIBLE PERMUTATION STATISTICS

be studied through noncommutative symmetric functions (using the internal product of Sym

[18, Section 5]) or quasisymmetric functions (using the internal coproduct of QSym [19]).

127

APPENDIX A

Summary of permutation statistics

The four tables in this appendix summarize the permutation statistics that appear in this

dissertation. For each statistic, we list the symbol used for the statistic, the name of the

statistic, the section where the statistic is defined, and new results (with the exception of

technical lemmas and propositions) in which the statistic appears.

We note that many of our results can be interpreted as being about statistics other than

the ones that appear in the result—for example, all three results about the peak number

listed below can also be interpreted as results about the valley number because these two

statistics are c-equivalent and equidistributed over Sn—but for each given statistic, we only

list results that explicitly mention the statistic.

We begin with a table of set-valued statistics.

Table 9. Set-valued permutation statistics

Statistic Name of Statistic Definition Results

Des descent set §1.1 Corollary 5.9

Pk peak set §1.2 Theorem 5.10

Val valley set §1.2

Lpk left peak set §1.2 Theorem 5.11

Rpk right peak set §1.2

Epk exterior peak set §1.2

Altdes alternating descent set §1.4

Desi,j partial descent sets §5.3.3 Theorem 5.15

128

CHAPTER A. SUMMARY OF PERMUTATION STATISTICS

Our next table lists integer-valued statistics.

Table 10. Integer-valued permutation statistics


inv inversion number §1.1

des descent number §1.2 Theorems 4.3, 4.5, 4.8, 4.12,4.16, 4.18, 5.28; Corollary 4.4

maj major index §1.2 Theorem 5.2

pk peak number §1.2 Theorems 3.8, 5.22;Corollary 3.10

val valley number §1.2

lpk left peak number §1.2 Theorem 5.24

rpk right peak number §1.2 Theorem 3.9; Corollary 3.10

epk exterior peak number §1.2 Theorem 3.11

dasc double ascent number §1.2 Theorem 3.12; Corollary 3.14

rdasc right double ascent number §1.2 Theorem 3.13

edasc exterior double ascent number §1.2 Theorem 3.13; Corollary 3.14

br number of biruns §1.2 Theorems 3.15, 5.30

udr number of up-down runs §1.2 Theorems 3.15, 4.16, 5.26;Corollary 4.21

lr long run statistic §1.2

lir long initial run statistic §1.2

lfr long final run statistic §1.2

sir short initial run statistic §1.2

sfr short final run statistic §1.2

altdes alternating descent number §1.4

as length of longest alternatingsubsequence

§3.5.4

imaj inverse major index §4.5

129


Next, we list joint statistics (i.e., ordered tuples). A few remarks are in order here. First,

we exclude the names and definitions of joint statistics, because these statistics have no special

names and their definitions come directly from the definitions of the individual statistics that

they are a tuple of. Second, we exclude a number of joint statistics which were mentioned

in our study of shuffle-compatible permutation statistics (Chapter 5) solely because they

are equivalent, r-equivalent, c-equivalent, or rc-equivalent to permutation statistics that we

study explicitly. (A list of such equivalences is given in Appendix B.) Finally, we exclude

joint statistics from Sections 5.5–5.6 that do not appear elsewhere in this dissertation.

Table 11. Joint permutation statistics

Statistic Results

(pk, des) Theorems 4.8, 5.21

(lpk, des) Theorems 4.12, 5.23

(udr, des) Theorem 5.25

(lpk, val, des) Theorems 4.18, 4.20

(Pk,Val) Theorems 3.3, 3.5, 5.29

(inv, pk) Corollary 4.11

(inv, pk, des) Theorem 4.9

(inv, lpk) Corollary 4.14

(inv, lpk, des) Theorem 4.13

(inv, lpk, val, des) Theorem 4.19

(des,maj) Theorem 5.27

Finally, we list type B permutation statistics, that is, statistics defined on signed permu-

tations.

130


Table 12. Type B permutation statistics


desB type B descent number §4.2 Corollaries 4.4, 4.7

fdes flag descent number §4.2 Corollaries 4.7, 4.21

neg number of negative letters §4.2

(neg, desB) Theorems 4.1, 4.3, 4.6, 4.15

(neg, fdes) Theorems 4.2, 4.5, 4.6, 4.20

131

APPENDIX B

Summary of permutation statistic equivalences

The following table gives a partial list of equivalences, r-equivalences, c-equivalences,

and rc-equivalences (defined in Sections 5.2.1–5.2.2) among permutation statistics that are

studied in this dissertation. Not all of these are explicitly proven in this dissertation, but the

proofs are very straightforward. We leave out some redundancies such as sir ∼c lir—omitted

since we include sir ∼ lir—as well as equivalences like (Lpk, val, des) ∼ (Lpk, br, des), which

is an immediate consequence of (Lpk, val) ∼ (Lpk, br).

Table 13. Equivalences among permutation statistics

Equivalences r-Equivalences

Des ∼ Altdes ∼ Lpk∪Val ∼ (Lpk,Val) Lpk ∼r Rpk

val ∼ epk lpk ∼r rpk

rpk ∼ epk sir ∼r lfr

rpk ∼ lr sfr ∼r lir

udr ∼ (lpk, val)

Epk ∼ (Epk, val) ∼ (Epk, udr) ∼ (Epk, br)

∼ (Lpk, val) ∼ (Lpk, udr) ∼ (Pk, udr)

c-Equivalences

Pk ∼c Val

sir ∼ lir ∼ Des1,0 pk ∼c val

sfr ∼ lfr ∼ Des0,1

(Pk, val) ∼ (Pk, br) rc-Equivalences

(Lpk, val) ∼ (Lpk, br) (pk, des) ∼rc (val, des)

(pk, val) ∼ (pk, br) ∼ (val, br) (lpk, val) ∼rc (lpk, pk)

132

APPENDIX C

Summary of shuffle-compatible permutation statistics

The following table summarizes every permutation statistic st that we know to be shuffle-

compatible, along with its shuffle algebra Ast and the dimension of the nth homogeneous

component of Ast. Here, Fn is the nth Fibonacci number defined by Fn := Fn−1 + Fn−2 for

n ≥ 3 and by F2 := 1 and F1 := 1.

Table 14. Shuffle-compatible permutation statistics

Statistic Shuffle Algebra Dimension of nthHomogeneous Component

Des, Altdes, Lpk∪Val,(Lpk,Val)

QSym 2n−1

des Theorem 5.28 n

maj Theorem 5.2(n2

)+ 1

(des,maj) Theorem 5.27(n3

)+ n

Pk,Val Algebra of peaks Π Fn

Lpk,Rpk Algebra of left peaks Π(`) Fn+1

pk, val, epk Theorem 5.22 b(n+ 1)/2clpk, rpk, lr Theorem 5.24 bn/2c+ 1

(pk, des), (val, des), (epk, des) Theorem 5.21 b(n+ 1)2/4c(lpk, des), (rpk, des), (lr, des) Theorem 5.23 bn2/4c+ 1

udr, as, (lpk, val), (lpk, pk),(lpk, epk), (rpk, val), (rpk, pk),

(rpk, epk), (lr, val), (lr, pk),(lr, epk)

Theorem 5.26 n

133

CHAPTER C. SUMMARY OF SHUFFLE-COMPATIBLE PERMUTATION STATISTICS

(udr, des), (lpk, val, des),(lpk, epk, des), (lpk, pk, des)

Theorem 5.25(n2

)+ 1

Des1,0 Des0,1, sir, lir, sfr, lfr 2

Desi,j 2i+j (if i+ j ≤ n− 1)

Epk, (Epk, val), (Epk, udr),(Epk, br), (Lpk, val),(Lpk, udr), (Pk, udr)

[27] Fn+2 − 1

134

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