Log-concave poset inequalities

TL;DR

This paper establishes new log-concavity inequalities for various set systems like matroids and greedoids, extending Mason conjectures and employing combinatorial linear algebra techniques.

Contribution

It introduces combinatorial proofs of log-concavity inequalities for set systems, extending previous results and establishing equality conditions using linear algebra and greedoid language.

Findings

Proved log-concavity inequalities for weighted feasible words.

Extended Mason conjectures to broader classes of set systems.

Rederived and extended Stanley's inequality and equality conditions.

Abstract

We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concavity inequalities for counting certain weighted feasible words, which generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. Notably, we prove matching equality conditions for both earlier inequalities and our extensions. In contrast with much of the previous work, our proofs are combinatorial and employ nothing but linear algebra. We use the language formulation of greedoids which allows a linear algebraic setup, which in turn can be analyzed recursively. The underlying non-commutative nature of matrices associated with greedoids allows us to proceed beyond polymatroids and prove the equality conditions. As further application of our tools, we rederive both Stanley's inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.