Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are $��$-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as $h$-polynomials of certain flag simplicial polytopes and which admit interesting Schur $��$-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 $��$-positivity in geometric combinatorics.

Final version; minor changes