487 research outputs found
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
Exponential Dowling structures
The notion of exponential Dowling structures is introduced, generalizing
Stanley's original theory of exponential structures. Enumerative theory is
developed to determine the M\"obius function of exponential Dowling structures,
including a restriction of these structures to elements whose types satisfy a
semigroup condition. Stanley's study of permutations associated with
exponential structures leads to a similar vein of study for exponential Dowling
structures. In particular, for the extended r-divisible partition lattice we
show the M\"obius function is, up to a sign, the number of permutations in the
symmetric group on rn+k elements having descent set {r, 2r, ..., nr}. Using
Wachs' original EL-labeling of the r-divisible partition lattice, the extended
r-divisible partition lattice is shown to be EL-shellable.Comment: 17 page
Inequalities for the h- and flag h-vectors of geometric lattices
We prove that the order complex of a geometric lattice has a convex ear
decomposition. As a consequence, if D(L) is the order complex of a rank (r+1)
geometric lattice L, then for all i \leq r/2 the h-vector of D(L) satisfies
h(i-1) \leq h(i) and h(i) \leq h(r-i).
We also obtain several inequalities for the flag h-vector of D(L) by
analyzing the weak Bruhat order of the symmetric group. As an application, we
obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of
geometric lattices which minimize Whitney numbers of the second kind. In
addition, we are able to give a combinatorial flag h-vector proof of h(i-1)
\leq h(i) when i \leq (2/7)(r + 5/2).Comment: 15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was
added regarding a solution to problem 4.
Cohomology of Dowling Lattices and Lie (Super)Algebras
AbstractWe extend a well-known relationship between the representation of the symmetric group on the homology of the partition lattice and the free Lie algebra to Dowling lattices
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