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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 Bettinumbers related to an N-graded minimal free resolution of F(M) are determinedby the Betti numbers related to the blocks of M. Similarly, we show that thehigher weight hierarchy of M is determined by the weight hierarchies of theblocks, as well. Drawing on these results, we show that when M is the cyclematroid of a cactus graph, the Betti numbers determine the higher weighthierarchy -- and vice versa. Finally, we demonstrate by way of counterexamplesthat this fails to hold for outerplanar graphs in general.

Betti numbers associated to the facet ideal of a matroid