Séminaire Lotharingien de Combinatoire, 84B.36 (2020), 12 pp.
Qiong Qiong Pan and Jiang Zeng
The γ-Coefficients of Brändén's (p,q)-Eulerian Polynomials and André Permutations
Abstract.
In 2008 Brändén proved a (p,q)-analogue of the
γ-expansion formula for
Eulerian polynomials and conjectured the divisibility of
the γ-coefficient
γn,k(p,q) by (p+q)k. As a follow-up, in 2012
Shin and Zeng showed that the fraction γn,k(p,q)/(p+q)k is a polynomial in
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.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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