  The Tutte polynomial for matroids is not directly applicable to polymatroids.
For instance, deletion-contraction properties do not hold. We construct a
polynomial for polymatroids which behaves similarly to the Tutte polynomial of
a matroid, and in fact contains the same information as the Tutte polynomial
when we restrict to matroids. This polynomial is constructed using lattice
point counts in the Minkowski sum of the base polytope of a polymatroid and
scaled copies of the standard simplex. We also show that, in the matroid case,
our polynomial has coefficients of alternating sign, with a combinatorial
interpretation closely tied to the Dawson partition.
