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 MM. Our main result is that these polynomials are determined by Betti numbers associated with N0N0-graded minimal free resolutions of the Stanley–Reisner ideals of MM and so-called elongations of MM. Generalizing Greene’s theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.