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 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.Comment: 21 page
To a matroid M with n edges, we associate the so-called facet ideal F(M)
generated by monomials corresponding to bases of M. We show that the Betti
numbers related to an N-graded minimal free resolution of F(M) are determined
by the Betti numbers related to the blocks of M. Similarly, we show that the
higher weight hierarchy of M is determined by the weight hierarchies of the
blocks, as well. Drawing on these results, we show that when M is the cycle
matroid of a cactus graph, the Betti numbers determine the higher weight
hierarchy -- and vice versa. Finally, we demonstrate by way of counterexamples
that this fails to hold for outerplanar graphs in general.Comment: 18 page
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