The ACO Seminar (2024–2025)

The h-vector of a matroid M is an important invariant related to the independence complex of M, which can also be covered as an evaluation of its Tutte polynomial. A well-known conjecture of Stanley posits that the h-vector of a matroid is a "pure O-sequence", meaning that it can be recovered by counting faces of a pure multicomplex. Merino has established Stanley's conjecture for the case of cographic matroids via a connection to chip-firing on graphs and the concept of a G-parking function. Inspired by these constructions, we introduce the notion of a cycle system for a matroid M . This leads to a collection of integer sequences that we call "parking functions" for M, which we show are in bijection with the set of bases of M. We study maximal coparking functions, and also how cycle systems behave under deletion and contraction. This leads to a proof of Stanley’s conjecture for the case of matroids that admit cycle systems (which for instance includes graphic matroids of K_{3,3}-minor free graphs). This is joint work with Scott Cory, Solis McClain, and David Perkinson.