In 2008 Brändén proved a $(p,q)$-analogue of the γ-expansion formula for Eulerian polynomials and conjectured the divisibility of the γ-coefficient ${\gamma }_{n,k}(p,q)$ by ${(p+q)}^k$. As a follow-up, in 2012 Shin and Zeng showed that the fraction ${\gamma }_{n,k}(p,q)/{(p+q)}^k$ is a polynomial in $\mathbb{N}[p,q]$. The aim of this paper is to give a combinatorial interpretation of the latter polynomial in terms of André permutations, a class of objects first defined and studied by Foata, Schützenberger and Strehl in the 1970s. It turns out that our result provides an answer to a recent open problem of Han, which was the impetus of this paper.

在2008年，Brändén证明了 $（p，q）$多项式的γ展开式的模拟，并推测了γ系数的可除性${\gamma }_{ñ，ķ}（p，q）$ 经过 ${（p+q）}^{ķ}$。作为后续措施，Shin和Zeng在2012年表明${\gamma }_{ñ，ķ}（p，q）/{（p+q）}^{ķ}$ 是...的多项式 $\mathbb{ñ}[p，q]$。本文的目的是根据André置换给出后一种多项式的组合解释，André置换是1970年代由Foata，Schützenberger和Strehl首次定义和研究的一类对象。事实证明，我们的结果为最近出现的汉族问题提供了答案，这是本文的推动力。