Binomial Eulerian polynomials for colored permutations

Abstract

Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are γ-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as h-polynomials of certain flag simplicial polytopes and which admit interesting Schur γ-positive symmetric function generalizations. This paper introduces analogues of these polynomials for r-colored permutations with similar properties and uncovers some new instances of equivariant γ-positivity in geometric combinatorics.