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May 11, 2026
Monday, May 11
Geometry and Combinatorics
Time: 15:30
Speaker: Anton Dochtermann (Texas State University)
Title: "Cycle systems and h-vectors of matroids"
Room: MC 108

Abstract: The h-vector of a matroid is an important invariant that has been the subject of intense study in recent years. A still open conjecture of Stanley posits that the h-vector of a matroid is a pure O-sequence, meaning that it can be obtained by counting faces of a pure multicomplex. Merino established Stanley's conjecture for the case of cographic matroids via chip-firing on graphs and the notion of a G-parking function. Inspired by these constructions, we introduce the notion of a "cycle system" for a matroid M - a family of circuits of M with overlap properties that mimic cut-sets in a graph. A choice of cycle system on M defines a collection of integer sequences that we call "coparking functions", which we show are in bijection with the set of bases of M. This leads to a proof of Stanley's conjecture for the case of matroids that admit cycle systems, which, for instance, include graphic matroids of cones as well as $K_{3,3}$-minor free graphs. Joint work with Scott Corry, Solis McClain, David Perkinson, and Lixing Yi.