The [formula omitted]-positivity of bivariate Eulerian polynomials via the Hetyei–Reiner action

Abstract

The bivariate Eulerian polynomials are defined by An(p,q)=∑π∈Snpodes(π)qedes(π),where odes(π) and edes(π) are the number of descents of permutation π in odd and even positions, respectively. In this paper, by the Hetyei–Reiner action, we show that for k≥1, the bivariate Eulerian polynomials A2k+1(p,q) and (1+p)−1A2k(p,q) are γ-positive, namely, they can be expressed in terms of the basis Bn≔{(pq)i(p+q)j(1+pq)n−2i−j|i,j≥0,2i+j≤n}with nonnegative coefficients.