Sign-Coherent Identities for Characteristic Polynomials of Matroids

Abstract

We derive an explicit formula for the difference χ(G;λ) − χ(G|X;λ)χ(Tx(G); λ)/(λ − 1), where χ(G;λ) is the characteristic polynomial of a simple matroid G, G|X is the restriction of G to a flat X in G, and Tx(G) is the complete principal truncation of G at the flat X. Two counting proofs of this formula are given. The first uses the critical problem and the second uses the broken-circuit complex. We also derive several inequalities involving Whitney numbers of the first kind and other numerical invariants.