$h$-vectors of small matroid complexes

TL;DR

This paper proves Stanley's conjecture for matroids of small rank and corank, and verifies it computationally for all matroids with up to nine elements, advancing understanding of matroid $h$-vectors.

Contribution

Provides simple constructive proofs for Stanley's conjecture in low-rank and corank cases and computational verification for small matroids.

Findings

Stanley's conjecture holds for matroids of rank ≤ 3 and corank 2.

Confirmed the conjecture for all matroids with up to nine elements using computer verification.

Established the conjecture as true in specific small cases, supporting its general validity.

Abstract

Stanley conjectured in 1977 that the $h$-vector of a matroid simplicial complex is a pure $O$-sequence. We give simple constructive proofs that the conjecture is true for matroids of rank less than or equal to 3, and corank 2. We used computers to verify that Stanley's conjecture holds for all matroids on at most nine elements.