Relaxed coparking functions and Stanley's conjecture for matroid $h$-vectors

Abstract

Stanley's conjecture asserts that the $h$-vector of a matroid is a pure $O$-sequence. Corry, Dochtermann, McClain, Perkinson and Yi introduced cycle systems, which give a bijective proof for the matroids that admit one, through coparking functions, and proposed generalized cycle systems. Every matroid admitting a cycle system is regular, and the Wagner and Petersen graphs admit no generalized cycle system consisting of circuits. We propose a relaxation. Where the coparking recursion breaks down, at the strata whose unique union is independent, the coparking functions are replaced by a fibre: a pure multicomplex with the $h$-vector of the dead node, the minor of the matroid attached to the stratum. We prove, for any matroid with a fixed basis, that whenever the required fibres exist the relaxed coparking functions form a pure multicomplex whose degree sequence is the $h$-vector of the matroid, so Stanley's conjecture follows. The proof rests on a version of Dhar's burning algorithm for a matroid with a fixed basis, which gives the purity, and on a deletion-contraction identity that computes the gap between the $h$-vector and the coparking functions as a sum of local $h$-vectors, one per dead node. Coned, biconed and triconed graphs carry a fibre system with their canonical spanning trees, as do the Wagner and Petersen graphs. Beyond graphs, every basis of a matroid of corank two, of a matroid of rank at most four or of a uniform matroid carries a fibre system, so Stanley's conjecture follows for those classes. We also give a graph of radius two on twelve vertices which, with its breadth-first spanning tree, carries no fibre system.