Characteristic polynomials of semimatroids and their connections to matroids, hyperplane arrangements and graph colorings

TL;DR

This paper explores the properties of characteristic polynomials of semimatroids, providing combinatorial interpretations, proving unimodality and log-concavity, and establishing connections to matroids, hyperplane arrangements, and graph colorings.

Contribution

It generalizes Whitney's Broken Circuit Theorem, extends the Rota-Heron-Welsh Conjecture to semimatroids, and introduces matroid assignments linking various combinatorial structures.

Findings

Coefficients of characteristic polynomials are unimodal and log-concave.

Provides a combinatorial interpretation of polynomial coefficients.

Establishes connections among semimatroids, hyperplane arrangements, and graph colorings.

Abstract

We primarily investigate the properties of characteristic polynomials of semimatroids. In particular, we provide a combinatorial interpretation of their coefficients, generalizing the Whitney's Broken Circuit Theorem. We also prove that the unsigned coefficients of the characteristic polynomial form a unimodal and log-concave sequence, extending the Rota-Heron-Welsh Conjecture to semimatroids. Furthermore, we present convolution identities for the multiplicative characteristic and Tutte polynomials of semimatroids using the M\"obius conjugation. Finally, motivated by Kochol's work, we introduce assigning matroids to establish connections among semimatroids, hyperplane arrangements, and graph colorings, with a particular focus on their characteristic polynomials.