Abstract. A long-standing conjecture of Stanley states that every CohenMacaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.
Abstract. Chen, Kitaev, Mütze, and Sun recently introduced the notion of universal partial words, a generalization of universal words and de Bruijn sequences. Universal partial words allow for a wild-card character ⋄, which is a placeholder for any letter in the alphabet. We extend results from the original paper and develop additional proof techniques to study these objects. For non-binary alphabets, we show that universal partial words have periodic ⋄ structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for an infinite family of universal partial words over non-binary alphabets.
The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper, we completely characterize the pairs (graph, matching complex) for which the matching complex is a homology manifold, with or without boundary. Except in dimension two, all of these manifolds are spheres or balls. §1. Introduction. The matching complex M(G) of a graph G is a simplicial complex representing the matchings (sets of independent edges) of the graph. There is an extensive literature describing the matching complexes of certain types of graphs. There are many results on the topology of the matching complexes of interesting classes of graphs. For example, there has been much study of chessboard complexes, m,n = M(K m,n). Björner et al. [2] prove that M(K m,n) is ν-connected, where ν = min{m, n, m+n+1 3 } − 2. Ziegler [20] shows that for m 2n − 1, M(K m,n) is shellable, and Jojić [9] uses this to give a recursion for the h-vectors of these chessboard complexes. Athanasiadis [1] studies vertex decomposability of skeleta of hypergraph matching complexes and chessboard complexes. Wachs [19] surveys results on the homology of chessboard complexes and matching complexes of the complete graph. Jonsson's dissertation (published as [10]) studies various complexes associated with graphs, including matching complexes. Kozlov [13] proves that, for ν n = n−2 3 , the matching complex M(P n+1) of the length n path is homotopy equivalent to the sphere S ν n when n mod 3 = 1, and the matching complex M(C n) of the n-cycle is homotopy equivalent to the sphere S ν n when n mod 3 = 0. (As is standard in graph theory, the subscript on a graph name indicates the number of vertices; for paths this is one more than the length.) Matching complexes of grid graphs have been studied by Braun and Hough [3] and by Matsushita [16]. Marietti and Testa [15] prove that matching complexes of forests are contractible or homotopy equivalent to a wedge of spheres. For caterpillar graphs, Jelić Milutinović et al. [8] give explicit formulas for the number of spheres in each dimension. They also study the connectivity of matching complexes of honeycomb graphs. We are interested in the reverse question: which simplicial complexes are matching complexes of graphs? In this paper, we will classify homology manifolds, with and without boundary, which are matching complexes. In §2, we review definitions and introduce several tools that we will rely on in later sections. In §3, we describe all graphs whose matching complexes are 1-and 2-dimensional spheres. In §4, we describe all homology manifolds
Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.
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