Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

The Rees product and cubical complexes

Tricia Muldoon Brown

Armstrong Atlantic State University

April 18, 2010

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Outline

Introduction

Results

Cubical Results

Rees multiple

Almost cubical posets

Questions

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Assumptions

P is a poset which is:

bounded below

graded, with rank function ρ.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Rees product

Definition

For two graded posets P and Q with rank function ρ the Reesproduct P ? Q, is the set of ordered pairs (p, q) in the Cartesianproduct P × Q with ρ(p) ≥ ρ(q). These pairs are partially orderedby (p, q) ≤ (p′, q′) if p ≤P p′, q ≤Q q′, andρ(p′)− ρ(p) ≥ ρ(q′)− ρ(q).

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

C4 ? C4

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Figure: The Rees product of two chains

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

C2

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Figure: The face lattice of the square, C2

Notation: Rees(P,Q) = ((P \ {0}) ? Q) ∪ {0, 1}

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

C2

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Figure: The face lattice of the square, C2

Notation: Rees(P,Q) = ((P \ {0}) ? Q) ∪ {0, 1}

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Rees(C2,C3)

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Figure: Rees(C2,C3)

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Observation

For P, a rank n poset, the Rees product Rees(P,Cn) is isomorphicto the Segre product ((P \ {0}) ◦ (Cn ? Cn)) ∪ {0, 1}.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Cohen-Macaulay

Theorem (Bjorner–Welker)

The Rees product of two Cohen-Macaulay posets isCohen-Macaulay.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Simplicial Mobius values

Theorem (Jonsson)

The Mobius function of the Rees product of the Boolean algebraBn on n elements with the n element chain Cn is given by the nthderangement number, that is,

µ(Rees(Bn,Cn)) = (−1)n+1 · Dn.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Cubical Mobius values

Theorem (Brown–Readdy)

The Mobius function of the Rees product of the face lattice of then-dimensional cube Cn with the n + 1 element chain Cn+1 is ntimes a signed derangement number, that is

µ(Rees(Cn,Cn+1)) = (−1)n · n · per

1 2 2 · · · 22 1 2 · · · 22 2 1 · · · 2...

......

. . ....

2 2 2 · · · 1

.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Simplicial poset

Definition

A poset P with a minimal element 0P is simplicial if the interval[0P , x ] is isomorphic to a Boolean algebra for all x ∈ P.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Corollary (Shareshian-Wachs)

Let P be a ranked simplicial poset of length n. Then

µ(Rees(P,Cn) =n∑

r=0

(−1)r−1Wr (P)r !.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Rees multiple

Definition

Let P be a rank n poset with rank function ρ and R = Rees(P,Cn).For any p ∈ P with ρ(p) = r , define the Rees multiple

xr ,p =r∑

j=1

µ(0R , (p, j))

Definition

A poset P is called lower uniform if [0P , x ] ' [0p, y ] whenρ(x) = ρ(y).

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Rees multiple

Definition

Let P be a rank n poset with rank function ρ and R = Rees(P,Cn).For any p ∈ P with ρ(p) = r , define the Rees multiple

xr ,p =r∑

j=1

µ(0R , (p, j))

Definition

A poset P is called lower uniform if [0P , x ] ' [0p, y ] whenρ(x) = ρ(y).

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Cubical poset

Definition

A poset P with a minimal element 0P is cubical if the interval[0P , x ] is isomorphic to the face lattice of a cube for all x ∈ P.

Proposition

Let P be a ranked cubical poset of length n. Then

µ(Rees(P,Cn) =n∑

r=0

(−1)r−1Wr (P)|xr |.

where xr is the Rees multiple for the cubical lattice.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Cubical poset

Definition

A poset P with a minimal element 0P is cubical if the interval[0P , x ] is isomorphic to the face lattice of a cube for all x ∈ P.

Proposition

Let P be a ranked cubical poset of length n. Then

µ(Rees(P,Cn) =n∑

r=0

(−1)r−1Wr (P)|xr |.

where xr is the Rees multiple for the cubical lattice.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Values

Rees multiple 0 1 2 3 4 5 6

Bn 1 1 2 6 24 120 720Cn 1 −1 2 −7 40

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Proposition

The rank r Rees multiple for the cubical lattice, xr , is given by therecursive formula

x0 = 1

xr = −r −r−1∑i=1

(r + 1− i)Wi (Cr−1)xi

where the ith Whitney number Wi (Cn) is the number of(i − 1)-dimensional faces in the n-dimensional cube.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Rees product as a Segre product

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W0(Cn)

W1(Cn)

W2(Cn) W2(Cn)

W3(Cn) W3(Cn) W3(Cn)

W4(Cn) W4(Cn) W4(Cn) W4(Cn)

Compute x4 by choosing any p ∈ Cn of rank 4.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Rees product as a Segre product

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W0(Cn)

W1(Cn)

W2(Cn) W2(Cn)

W3(Cn) W3(Cn) W3(Cn)

W4(Cn) W4(Cn) W4(Cn) W4(Cn)

Compute x4 by choosing any p ∈ Cn of rank 4.Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Example

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W0(C3)

W1(C3)

W2(C3) W2(C3)

W3(C3) W3(C3) W3(C3)

p p p p

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Example

x4 =4∑

i=1

µ(0, (p, i))

=4∑

i=1

− ∑0≤x<(p,i)

µ(0, x)

= −(2 ·W3(C3)x3 + 3 ·W2(C3)x2 + 4 ·W1(C3)x1 + 4W0(C3)x0)

= −(2 · 6(−7) + 3 · 12 · 2 + 4 · 8(−1) + 4 · 1 · 1) = 40

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Example

x4 =4∑

i=1

µ(0, (p, i))

=4∑

i=1

− ∑0≤x<(p,i)

µ(0, x)

= −(2 ·W3(C3)x3 + 3 ·W2(C3)x2 + 4 ·W1(C3)x1 + 4W0(C3)x0)

= −(2 · 6(−7) + 3 · 12 · 2 + 4 · 8(−1) + 4 · 1 · 1) = 40

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Example

x4 =4∑

i=1

µ(0, (p, i))

=4∑

i=1

− ∑0≤x<(p,i)

µ(0, x)

= −(2 ·W3(C3)x3 + 3 ·W2(C3)x2 + 4 ·W1(C3)x1 + 4W0(C3)x0)

= −(2 · 6(−7) + 3 · 12 · 2 + 4 · 8(−1) + 4 · 1 · 1) = 40

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Example

x4 =4∑

i=1

µ(0, (p, i))

=4∑

i=1

− ∑0≤x<(p,i)

µ(0, x)

= −(2 ·W3(C3)x3 + 3 ·W2(C3)x2 + 4 ·W1(C3)x1 + 4W0(C3)x0)

= −(2 · 6(−7) + 3 · 12 · 2 + 4 · 8(−1) + 4 · 1 · 1) = 40

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Non-recursive formula

Proposition

Let fn,k−1 = Wk(Cn). The rank r Rees multiple, xr , for the cubicallattice is given by∑

(−1)l(kl −kl−1 + 1) · · · (k2−k1 + 1)(k1 + 1)fkl ,kl−1· · · fk2,k1fk1,−1

where the sum is over all integers 0 < k1 < k2 < · · · < kl = r − 1where 1 ≤ l ≤ r .

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Simplicial case

Corollary

The number of permutations in the rth symmetric group, r !, isgiven by∑

(−1)l(kl − kl−1 + 1) · · · (k2 − k1 + 1)k1

(klkl−1

)· · ·(k2k1

)(k10

)where the sum is over all integers 0 < k1 < k2 < · · · < kl = r suchthat 1 ≤ l ≤ r .

This sum is over all l-compositions(kl − kl−1) + · · ·+ (k2 − k1) + k1 of r where 0 ≤ l ≤ r − 1

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Simplicial case

Corollary

The number of permutations in the rth symmetric group, r !, isgiven by∑

(−1)l(kl − kl−1 + 1) · · · (k2 − k1 + 1)k1

(klkl−1

)· · ·(k2k1

)(k10

)where the sum is over all integers 0 < k1 < k2 < · · · < kl = r suchthat 1 ≤ l ≤ r .

This sum is over all l-compositions(kl − kl−1) + · · ·+ (k2 − k1) + k1 of r where 0 ≤ l ≤ r − 1

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Almost cubical posets

Definition

Let Cn,k be the face poset of the cubical complex consisting of thethe boundaries of two n-dimensional cubes joined at ak-dimensional face adjoined with a maximal element.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Squares

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Figure: The boundaries of two squares joined at an edge

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

C2,1

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Figure: C2,1Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Mobius function

Corollary

The Mobius function, µ(Rees(Cn,k ,Cn+1)), is given by

2 · µ(Rees(Cn,Cn+1)) + (n − k − 1) · µ(Rees(Ck ,Ck+1)).

This follows from the fact

Wi (Cn,k) = 2Wi (Cn)−Wi (Ck)

for 0 ≤ i ≤ n.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Mobius function

Corollary

The Mobius function, µ(Rees(Cn,k ,Cn+1)), is given by

2 · µ(Rees(Cn,Cn+1)) + (n − k − 1) · µ(Rees(Ck ,Ck+1)).

This follows from the fact

Wi (Cn,k) = 2Wi (Cn)−Wi (Ck)

for 0 ≤ i ≤ n.

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Proposition

Let P and Q be ranked posets of length n. If

Wk(P) = Wk(Q) for all k = 0, 1, . . . n, and

µ([0P , x ]) = µ([0Q , y ]) where ρP(x) = ρQ(y),

thenµ(Rees(P,Cn)) = µ(Rees(Q,Cn))

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

More than 2 cubes

Corollary

Let C jn,k be the face poset of the cubical complex consisting of the

the boundaries of j n-dimensional cubes joined at a k-dimensionalface adjoined with a maximal element. Then the Mobius functionµ(Rees(C j

n,k ,Cn+1)) equals

j · µ(Rees(Cn,Cn+1)) + (j − 1) · (n − k − 1) · µ(Rees(Ck ,Ck+1)).

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Questions and Comments

1 Is the are cubical q-analogue?

2 Almost uniform?

3 What is the Rees multiple for other uniform or lower uniformposets? Does it have any meaning?

4 Similar results for t-ary tree.

5 Can we find |xr | as a sum of positive terms? What else does itcount?

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Questions and Comments

1 Is the are cubical q-analogue?

2 Almost uniform?

3 What is the Rees multiple for other uniform or lower uniformposets? Does it have any meaning?

4 Similar results for t-ary tree.

5 Can we find |xr | as a sum of positive terms? What else does itcount?

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Questions and Comments

1 Is the are cubical q-analogue?

2 Almost uniform?

3 What is the Rees multiple for other uniform or lower uniformposets? Does it have any meaning?

4 Similar results for t-ary tree.

5 Can we find |xr | as a sum of positive terms? What else does itcount?

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Questions and Comments

1 Is the are cubical q-analogue?

2 Almost uniform?

3 What is the Rees multiple for other uniform or lower uniformposets? Does it have any meaning?

4 Similar results for t-ary tree.

5 Can we find |xr | as a sum of positive terms? What else does itcount?

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Questions and Comments

1 Is the are cubical q-analogue?

2 Almost uniform?

3 What is the Rees multiple for other uniform or lower uniformposets? Does it have any meaning?

4 Similar results for t-ary tree.

5 Can we find |xr | as a sum of positive terms? What else does itcount?

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University

Introduction Results Cubical results Rees multiple Almost cubical posets Conclusion

Conclusion

The End

Tricia Muldoon Brown: The Rees product and cubical complexes Armstrong Atlantic State University