Discrete CATS Seminar
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U N I V E R S I T Y O F K E N T U C K Y

DISCRETE
CATS
SEMINAR

WHERE CATS =
COMBINATORICS,
ALGEBRA,
TOPOLOGY
&
STATISTICS!

845 PATTERSON OFFICE TOWER

2008 - 2009

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DOCTORAL DEFENSE

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"Rees products of posets and inequalities"

Patricia Muldoon Brown

University of Kentucky

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Tuesday, April 14, 2009

3:00 pm, 745 Patterson Office Tower

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Abstract:

In this talk we will look at properties of two different posets using
different combinatorial perspectives: (i) the Rees product of the face
lattice of the n-dimensional cube with the chain and
(ii) the partition
lattice.

Recently Björner and Welker defined a poset product called the Rees
product which is motivated by the Rees algebra of commutative algebra.
They showed that the Rees product of two Cohen-Macaulay posets
preserves the Cohen-Macaulay property. Very few examples of Rees
products of Cohen-Macaulay posets have been studied. Jonsson showed
the Möbius function of the Rees product of the Boolean algebra with
the chain is a derangement number, solving an open problem of Björner
and Welker. We show the Möbius function of the Rees product of the
face lattice of the n-dimensional cube with the chain is given by n
times a signed derangement number. One of the proofs we give is
bijective between barred signed permutations which label the maximal
chains of this poset and a set of labeled augmented skew diagrams. As
a corollary, we give a bijective proof of Jonsson's result. We then
use the labeled diagrams to index a basis for the homology of the
order complex of the Rees product of the n-cube with the chain and
determine a representation of the homology of its order complex over
the symmetric group.

It is a classical result due to many authors, including Niven, de
Bruijn and Viennot, that the alternating descent words abab...ab and
baba...ba maximize the number of permutations having a particular
descent pattern. Motivated by this work and the general program of
determining all the flag vector inequalities among n-dimensional
polytopes and more general manifolds, we study flag vector
inequalities for the Dowling lattice, especially the special case of
the partition lattice. Ehrenborg and Readdy conjectured that the
maximum flag h-vector entry for the partition lattice occurs for the
almost alternating descent words baba...bab or baba...babb. We show
the flag vector of the partition lattice can be determined using a
weighted boustrophedon transformation and determine many of the flag
vector inequalities which hold among its flag vector entries. We end
with a number of conjectures.