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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
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