Poset Convex-ear Decompositions and Applications to the Flag H-vector

Possibly the most fundamental combinatorial invariant associated to a finite simplicial complex is its f-vector, the integral sequence expressing the number of faces of the complex in each dimension. The h-vector of a complex is obtained by applying a simple invertible transformation to its f-vector, and thus the two contain the same information. Because some properties of the f-vector are easier expressed after applying this transformation , the h-vector has been the subject of much study in geometric and algebraic combinatorics. A convex-ear decomposition, first introduced by Chari in [7], is a way of writing a simplicial complex as a union of subcomplexes of simplicial polytope boundaries. When a (d − 1)-dimensional complex admits such a decomposition, its h-vector satisfies, for i < d/2, h i ≤ h i+1 and h i ≤ h d−i. Furthermore, its g-vector is an M-vector. We give convex-ear decompositions for the order complexes of rank-selected sub-posets of supersolvable lattices with nowhere-zero Möbius functions, rank-selected sub-posets of geometric lattices, and rank-selected face posets of shellable complexes (when the rank-selection does not include the maximal rank). Using these decompositions, we are able to show inequalities for the flag h-vectors of supersolvable lattices and face posets of Cohen-Macaulay complexes. Finally, we turn our attention to the h-vectors of lattice path matroids. A lattice path matroid is a certain type of transversal matroid whose bases correspond to planar lattice paths. We verify a conjecture of Stanley in the special case of lattice path matroids and, in doing so, introduce an interesting new class of monomial order ideals. relocation, he and his family finally settled in the Washington, D.C. suburb of Hern-don, Virginia. Jay attended Oakton High School, and later enrolled in nearby George Mason University as a philosophy major. Two years and many Aristotle treatises later, Jay switched to a mathematics concentration. After obtaining a bachelor's degree, Jay worked several odd jobs in the Northern Virginia area. He eventually entered the mathematics Ph.D. program at Cornell University in Ithaca, New York, initially as a topologist. Six short years later, his advisor suggested that he graduate. After graduation, Jay will be a Robert D. Adams visiting assistant professor at the University of Kansas.