Ph.D. (2008) Cornell University
Abstract: 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 more easily 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, 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, hi ≤ hi + 1 and hi ≤ hd – i. Furthermore, its g-vector is an M-vector.
We give convex-ear decompositions for the order complexes of rank-selected subposets of supersolvable lattices with nowhere-zero Möbius functions, rank-selected subposets 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.