Matroid invariants arising from matroidal mixed Eulerian numbers

Determine which matroid invariants arise as matroidal mixed Eulerian numbers, namely as the intersection numbers obtained by pairing matroid classes on the permutohedral variety with monomials in the hypersimplex divisors.

Background

Matroidal mixed Eulerian numbers are intersection numbers of a matroid class [X_M] in the Chow ring of the permutohedral variety with products of the divisors L_1, ..., L_n associated with hypersimplices. They generalize the classical mixed Eulerian numbers and encode information about the matroid through intersection theory.

The problem is posed as a question from Berget, Spink, and Tseng concerning the range of matroid invariants captured by these intersection numbers. The paper subsequently resolves the question by proving that the complete collection of matroidal mixed Eulerian numbers is equivalent to Derksen’s G-invariant for loopless matroids; nevertheless, the quoted passage explicitly identifies the question as open before that resolution.