Equality cases of the Stanley--Yan log-concave matroid inequality
Swee Hong Chan, Igor Pak
TL;DR
This work settles Stanley's problem on equality cases for the Stanley--Yan log-concave inequality in matroids. It delivers a complete, explicit description of equality cases for k=0 and shows that for k≥1 the equality problem is not likely to admit a concise description within the polynomial hierarchy, unless PH collapses. The authors develop a dual approach: a positive, atlas-based construction that characterizes equality for k=0, and a negative, complexity-theoretic framework that reduces to spanning-tree counting and related graph-theoretic encodings to demonstrate hardness for k≥1. Key technical tools include combinatorial atlases with hyperbolic matrices, continued fractions linked to planar graphs and spanning trees, and discrete polymatroid vanishing criteria. The results illuminate a stark contrast between k=0 and k≥1 cases and open new directions at the intersection of matroid theory, complexity, and algebraic combinatorics.
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.