Equality cases of the Stanley--Yan log-concave matroid inequality

TL;DR

This paper characterizes when the Stanley--Yan log-concave matroid inequality holds with equality, extending previous results to all matroids and exploring the complexity of equality cases for larger intersections.

Contribution

It provides a complete description of equality cases for the Stanley--Yan inequality for all matroids and intersection sizes, resolving a long-standing open problem.

Findings

Explicit equality conditions for k=0 extend Stanley's results to all matroids.

For k≥1, equality cases are computationally complex and unlikely to be fully characterized.

The results deepen understanding of matroid base distributions and log-concavity properties.

Abstract

The \emph{Stanley--Yan} (SY) \emph{inequality} gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with $k$ fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY~inequality, and proved total equality conditions for regular matroids in the case $k=0$. In this paper, we completely resolve Stanley's problem. First, we obtain an explicit description of the equality cases of the SY inequality for $k=0$, extending Stanley's results to general matroids and removing the ``total equality'' assumption. Second, for $k\ge 1$, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level.