Action graphs and Catalan numbers - …jeb2md/Claremont.pdf · Action graphs and Catalan numbers...

Action graphs and Catalan numbers

Julie Bergner

University of California, Riverside

March 4, 2015

Julie Bergner Action graphs and Catalan numbers

Catalan numbers

The Catalan numbers are a sequence of natural numbers

C0,C1,C2, . . .

given by the recursive formula

C0 = 1

Ck+1 =k∑

i=0

CiCk−i .

So, the first few Catalan numbers are:

C0 = 1

C1 = C0C0 = 1

C2 = C0C1 + C1C0 = 2

C3 = C0C2 + C1C1 + C2C0 = 5

C4 = C0C3 + C1C2 + C2C1 + C3C0 = 14.


Where do Catalan numbers come from?

They are known to count at least 207 different kinds ofcombinatorial objects!

Ck is the number of ways of parenthesizing k + 1 variablespairwise.

(a)

(ab)

(ab)c a(bc)

(ab)(cd) a((bc)d) a(b(cd)) ((ab)c)d (a(bc))d

Ck is the number of planar rooted trees with k edges.


Directed graphs

Our goal is to give a new way to get the Catalan numbers, using acertain family of directed graphs.

Recall that a directed graph has vertices and edges, and the edgeshave a direction. We denote them by an arrow.

• // • //

��

•

• //

??~~~~~~~•

A path in a directed graph is a sequence of edges with compatibledirection.

In this talk, we’ll include “trivial paths”, which stay at a singlevertex.


Labeled directed graphs

We will also assume that every vertex of our directed graph islabeled by a natural number.

•0 // •1 //

��

•3

•2 //

>>||||||||•3


Action graphs

In recent work with Hackney on group actions on groups, we founda sequence of labeled directed graphs which begins as follows.

A0 : •0 A1 : •0 // •1

A2 : A3 : •3 •3

•2 •2 •2

OO

•2

OO

•0

OO

// •1

OO

•3 •0

OO

��

oo // •1

OO

��•3 •3


How are these graphs defined?

Begin with the graph A0 with one vertex, labeled by 0, and noedges.

Inductively, we build Ak+1 from Ak by looking at paths in Ak .

For every directed path in Ak , add a new edge starting at thesource of that path and ending at a a new vertex labeled by k + 1.


Counting new vertices

How many new vertices does each graph Ak have?

Theorem (Alvarez-B-Lopez, Definition 208?)

The number of new vertices in Ak is given by Ck .

The idea is the proof is to see that the edges coming out of thevertex labeled by 1 are formed just as the arrows coming out of thevertex labeled by 0 at the previous step, and so forth.


Comparison with planar rooted trees

Often with new approaches to the Catalan numbers, it is nice toconstruct a direct comparison to one of the other known ways toobtain them.

Theorem (Alvarez-B-Lopez)

There is a one-to-one correspondence between new vertices in Ak

and planar rooted trees with k edges.

The idea is to assemble the planar rooted trees in such a way thatwe obtain the action graph.


But where did action graphs come from?

In joint work with Hackney, we wanted to find a diagrammaticapproach to group actions on other groups.

Let’s start by looking at what this would mean for groups.

Consider the following labeled directed graphs:

•0 •0 //•1 •0 //•1 //•2 · · ·

Let [n] = (•0 → · · · → •n).

Assign to each [n] a set Xn.


We want to impose the following conditions:

X0 = ∗, a set with one element.

X1 is any set.

X2∼= X1 × X1.

More generally, Xn∼= X1 × · · · × X1︸︷︷︸

n

.

But what does this structure tell us?


Start with the set X1.

If X2∼= X1 × X1, we can “compose” or “multiply” elements in X1

with one another.

There is a map X0 → X1 which specifies an “identity element”.

The fact that X3∼= X1 × X1 × X1 tells us that the composition is

associative.

Thus, we get a monoid structure.

Can also incorporate inverses with an additional condition.


Groups acting on groups

Recall that a group G acts on a set X if there exists a function

G × X → X

satisfying some conditions.

The idea is that an element of the group takes each element of theset to another element of the set.

Here, we want to assume that X is also a group, so that we have agroup acting on another group.

We’d like to find a way to describe this kind of structure viadiagrams.


Consider the following two directed graphs.

1 y z

∗ 0

p

OO

wg //

p•gOO

x

p•idx

OO

The first acts on the second.

We can define two families of graphs, where the first acts on thesecond.

Some of the graphs in the second family give the action graphs.


Further work

There are other algebraic structures that can be described bydiagrams: abelian monoids/groups, categories, operads.

These structures are of interest up to homotopy, when welook at diagrams of spaces.

Each of these structures can also be considered with a groupaction.

Are there more examples?

Do the diagrams we get have other interesting patterns?


References

Julia E. Bergner and Philip Hackney, Reedy categories whichencode the notion of category actions, to appear in Fund.Math., preprint available at math.AT/1207.3467.

Gerardo Alvarez, Julia E. Bergner, and Ruben Lopez, Actiongraphs and Catalan numbers, preprint available atmath.CO/1503.00044.


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