Internally Perfect Matroids

TL;DR

This paper introduces the class of internally perfect matroids, proves they satisfy Stanley's conjecture on the $h$-vector being a pure $ ext{O}$-sequence, and explores their properties and minors.

Contribution

It defines internally perfect matroids using Las Vergnas' internal order and proves they satisfy Stanley's conjecture, expanding the classes of matroids known to do so.

Findings

Internally perfect matroids satisfy Stanley's conjecture.

Every minor of an internally perfect matroid is internally perfect under certain conditions.

Internally perfect matroids form a new class with desirable algebraic properties.

Abstract

In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restriction on deletions, every minor of an internally perfect ordered matroid is internally perfect.