The representation of Sₙ on the cohomology of the permutohedral variety and gamma vectors of partitioned permutohedra

Foata and Sch\"{u}tzenberger gave an expansion for the Eulerian polynomial Aₙ(t) in terms of the basis ʲ(1+t)ⁿ⁻¹⁻²ʲ\ for the space of polynomials $f(t)$ satisfying f(t)=tⁿ⁻¹f(1/t). We generalize this result in two ways. First, we provide an analogue for the graded representation of the symmetric group Sₙ on the cohomology of the permutohedral variety. Then we give expansions h-polynomials of polytopes obtained by cutting permutohedra with hyperplanes orthogonal to simple roots in terms of the same basis.