The consecutive pattern poset is the infinite partially ordered set of all permutations where σ ≤ τ if τ has a subsequence of adjacent entries in the same relative order as the entries of σ. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and ...

In this paper we give convex-ear decompositions for the order complexes of several classes of posets, namely supersolvable lattices with non-zero Möbius functions and rank-selected subposets of such lattices, rank-selected geometric lattices, and rank-selected face posets of shellable complexes which do not include the top rank. These decompositions give us many new inequalities for the h-vecto...

Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. complex expose large amount information helpful way, but are not easy to construct, often requiring deep about the structure complex. It is natural ask whether shellings may be efficiently found computationally. In recent paper, Goaoc, Pat\'ak, Pat\'akov\'a, Tancer Wagner gave negative ans...

We develop an iterated homology theory for simplicial complexes. This theory is a variation on one due to Kalai. For 1 a simplicial complex of dimension d − 1, and each r = 0, . . . , d , we define r th iterated homology groups of 1. When r = 0, this corresponds to ordinary homology. If 1 is a cone over 1′, then when r = 1, we get the homology of 1′. If a simplicial complex is (nonpure) shellab...

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal ...

Motivated by the work of Salvetti and Settepanella [24, Combinatorial Morse theory and minimality of hyperplane arrangements, Remark 4.5], we give a purely combinatorial description of a class of discrete Morse functions having a minimal number of critical cells for the Salvetti complex of any linear arrangement. We start by studying certain total orderings of the cells of shellable regular CW-...