The $γ$-coefficients of Branden's $(p,q)$-Eulerian polynomials and André permutations

Qiong Qiong Pan, Jiang Zeng

Published 2019-10-03Version 1

P. Br\"and\'en (European J. Combin. 29 (2008), no. 2, 514--531) studied a $(p,q)$-analogue of the classical Eulerian polynomials $A_n(p,q,t)$ and conjectured that its $\gamma$-coefficient $a_{n,k}(p,q)$ is divisible by $(p+q)^k$. The aim of this paper is to show that the quotient $d_{n,k}(p,q):=a_{n,k}(p,q)/(p+q)^k$ is the enumerative polynomial of Andr\'e permutations of the second kind of size $n$ with $k$ descents. In addition, our result leads to a combinatorial model for G.-N. Han's recent $q$-Euler numbers (arXiv:1906.00103v1).