Pure O-sequences and matroid h-vectors

TL;DR

This paper advances the understanding of matroid h-vectors by proposing a new approach centered on pure O-sequences, proving Stanley's conjecture for rank 3 matroids and discussing potential generalizations.

Contribution

It introduces a novel abstract method focusing on pure O-sequences, proves Stanley's conjecture for rank 3 matroids, and offers a new perspective for tackling the conjecture in general.

Findings

Proved Stanley's conjecture for rank 3 matroids.

Proposed and settled a conjecture on pure O-sequences in small socle degrees.

Discussed a potential approach for the general case of Stanley's conjecture.

Abstract

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators.