Characteristic polynomials of semimatroids and their connections to matroids, 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.