In 1977 Stanley conjectured that the h-vector of a matroid independence complex is a pure O-sequence. In this paper we use lexicographic shellability for matroids to motivate a new approach to proving Stanley's conjecture. This suggests that a pure O-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank d matroids on at most 2d elements, then it holds for all matroids of rank d.
Abstract. We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac.
We prove that the external activity complex Act < (M ) of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order < ext/int on M provides a shelling of Act < (M ). We also show that every linear extension of LasVergnas's internal order < int on M provides a shelling of the independence complex IN (M ). As a corollary, Act < (M ) and M have the same h-vector. We prove that, after removing its cone points, the external activity complex is contractible if M contains U 3,1 as a minor, and a sphere otherwise.
We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality.Further, for C 2 -convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.