A generalization of weight polynomials to matroids

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M . Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M . Generalizing Greene’s theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.