  It is well known that the Eulerian polynomial is the Hilbert series of the
cohomology of the permutahedral variety. Stanley obtained a formula showing
that the cohomology carries a permutation representation of $\mathfrak{S}_n$.
We answer a question of Stembridge on finding an explicit permutation basis of
this cohomology. We observe that the Feichtner-Yuzvinsky basis for the Chow
ring of the Boolean matroid is such a permutation basis, and then we construct
an $\mathfrak{S}_n$-equivariant bijection between this basis and codes
introduced by Stembridge, thereby giving a combinatorial proof of Stanley's
formula. We obtain an analogous result for the stellahedral variety. We find a
permutation basis of the permutation representation its cohomology carries.
This involves the augmented Chow ring of a matroid introduced by Braden, Huh,
Matherne, Proudfoot and Wang. Along the way, we obtain a general result on
augmented Chow rings (which was also independently obtained by Eur) asserting
that augmented Chow rings of matroids are actually Chow rings in the sense of
Feichtner and Yuzvinsky. In the last part of the paper, we study enumerative
aspects of the permutahedra and the stellohedra related to these permutation
bases.
