341 research outputs found
A second look at the toric h-polynomial of a cubical complex
We provide an explicit formula for the toric -contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the -polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction
Cyclotomic factors of the descent set polynomial
We introduce the notion of the descent set polynomial as an alternative way
of encoding the sizes of descent classes of permutations. Descent set
polynomials exhibit interesting factorization patterns. We explore the question
of when particular cyclotomic factors divide these polynomials. As an instance
we deduce that the proportion of odd entries in the descent set statistics in
the symmetric group S_n only depends on the number on 1's in the binary
expansion of n. We observe similar properties for the signed descent set
statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up
notatio
Rees Products of Posets and Inequalities
In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric group both indexed by the set of labeled augmented skew diagrams. We also show that the Möbius function of the Rees product of a graded poset with the t-ary tree and the Rees product of its dual with the t-ary tree coincide. We discuss labelings for Rees and Segre products in general, particularly the Rees product of the face lattice of a polytope with the chain. We also look at cases where the Möbius function of a poset is equal to the permanent of a matrix and we consider local h-vectors for the barycentric subdivision of the n-cube. In each section we state open conjectures. The second poset in this dissertation is the Dowling lattice. In particular we look at the k = 1 case, that is, the partition lattice. We study inequalities on the flag vector of the partition lattice via a weighted boustrophedon transform and determine a more generalized version for the Dowling lattice. We generalize a determinantal formula of Niven and conclude with conjectures and avenues of study
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