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 hpolynomials of certain flag simplicial polytopes and which admit interesting Schur γpositive 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. Date: December 1, 2018.

Real-rootedness of variations of Eulerian polynomials

“…Recently, Gustafsson and Solus [17] proved that both d + n,r (z) and d − n,r (z) have only real roots, and Brändén and Solus [9] further proved that d + n,r (z) ≪ d − n,r (z). Recently, Athanasiadis [5] introduced a generalization A n,r (z) of A n (z) to the wreath product group Z r ≀ S n and further studied their symmetric function generalizations. The polynomial A n,r (z) is defined by the formula Athanasiadis [5] also studied the symmetric decomposition of A n,r (z) as…”

“…Recently, Athanasiadis [5] introduced a generalization A n,r (z) of A n (z) to the wreath product group Z r ≀ S n and further studied their symmetric function generalizations. The polynomial A n,r (z) is defined by the formula Athanasiadis [5] also studied the symmetric decomposition of A n,r (z) as…”

“…Gamma-positivity directly implies palindromicity and unimodality and appears widely in combinatorial and geometric contexts, see [4] for a survey. The following variation of Eulerian polynomials A n (z) := 1 + z n m=1 n m A m (z), first studied by Postnikov, Reiner, and Williams [24,Section 10.4], are also γ-positive and have attracted a lot of interest recently [26,23,5,22]. Shareshian and Wachs [26] called them binomial Eulerian polynomials and further studied a symmetric function generalization of them, which are shown to be equivariant γ-positive.…”

“…The first one is the real-rootedness of A n (z), which confirms Ma, Ma, and Yeh's conjecture. Another application is the real-rootedness of A + n,r (z) and A − n,r (z), the sum of which form the binomial Eulerian polynomials for colored permutations A n,r (z), defined by Athanasiadis [5] recently. The polynomials A n (z) and A + n,r (z) can be interpreted as h-polynomials of boundary complexes of certain simplicial polytopes, see [5,24].…”

“…Another application is the real-rootedness of A + n,r (z) and A − n,r (z), the sum of which form the binomial Eulerian polynomials for colored permutations A n,r (z), defined by Athanasiadis [5] recently. The polynomials A n (z) and A + n,r (z) can be interpreted as h-polynomials of boundary complexes of certain simplicial polytopes, see [5,24]. In our third application, the polynomials E s n (z) for certain s-inversion sequences are shown to be such h-polynomials, which are related to the edgewise subdivisions of simplexes.…”