The combinatorial Chow ring of a matroid produces log-concave sequences from a list of polytopes called "generalized permutahedra". If we take S_n-invariant polytopes, then we obtain matroid invariants, but what invariants do we obtain? We will discuss how many such invariants are linear combinations of the h-vector of the independence complex of a matroid by "mixed Eulerian numbers", and how this proves a strengthening of a conjecture of Dawson on the log-concavity of the matroid h-vector.