A multiplication identity for characteristic polynomials of matroids

The main result in this paper is the following multiplication identity for the characteristic polynomial chi (G; lambda) of a rank-r matroid G: chi (G; lambdaxi) = Sigma(X: Xis an element ofL(G)) chi (G/X;lambda)xi(r-rank(G\X))chi(G\X;xi), where the sum ranges over all flats of G, the matroid G\X is the restriction of G to X, and the matroid G/X is the contraction of G by X. We give three proofs. One proof is algebraic-and applies to all matroids. The other two proofs, less general but giving more insight and meaning, are, counting arguments based on the critical problem for representable matroids and the coloring problem for graphs. (C) 2003 Elsevier Inc. All rights reserved.