Interpretations of the Tutte and characteristic polynomials of matroids

Abstract

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 $$X\subseteq E$$ is called $$(M,<)$$ -compatible if $$X\cap C\ne \{\min (C)\}$$ for each circuit C of M. We show that the Tutte polynomial of M equals $$\sum x^{r(M/X)}y^{r^*(M|X)}$$ where X runs through the subsets of E such that X and $$E{{\setminus }}X$$ are $$(M^*,<)$$ - and $$(M,<)$$ -compatible, respectively. Similarly, the characteristic polynomial of M equals $$\sum (-1)^{|X|}(k-1)^{r(M/X)}$$ where X runs either through $$(M^*,<)$$ -compatible subsets of E, or through the independent sets of M such that X and $$E{{\setminus }}X$$ are $$(M^*,<)$$ - and $$(M,<)$$ -compatible, respectively.