Interpretations of the Tutte and characteristic polynomials of matroids

We study interpretations of the Tutte and characteristic polynomials of matroids. If M is a matroid with rank function r whose ground set E is given with a linear ordering <, then X. E is called ( M,<)-compatible if X n C = {min(C)} for each circuit C of M. We show that the Tutte polynomial of M equals xr ( M/ X) yr * ( M|X) where X runs through the subsets of E such that X and E\ X are (M*,<)- and ( M,<)-compatible, respectively. Similarly, the characteristic polynomial of M equals (-1)| X| (k - 1)r ( M/ X) where X runs either through ( M *,<)-compatible subsets of E, or through the independent sets of M such that X and E\ X are ( M *,<)- and ( M,<)-compatible, respectively.