Lexicographic shellability, matroids and pure order ideals

TL;DR

This paper explores a combinatorial approach to Stanley's conjecture on matroid $h$-vectors, proving it for matroids of rank up to four and establishing a reduction to smaller cases.

Contribution

It introduces a strengthened conjecture based on lexicographic shellability and proves it for low-rank matroids, advancing understanding of $h$-vectors and pure $O$-sequences.

Findings

Proves Stanley's conjecture for matroids of rank at most four.

Establishes a reduction principle linking conjecture validity to smaller matroids.

Suggests a combinatorial construction of pure $O$-sequences from shelling data.

Abstract

In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This suggests that a pure $O$-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank $d$ matroids on at most $2d$ elements, then it holds for all matroids.