Brändén's (p,q)-Eulerian polynomials, André permutations and continued fractions

2019/10/03 by Qiong Qiong Pan, Jiang Zeng, Pan, Qiong Qiong +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1910.01747

openalex publication_date 2019/10/03 · arxiv created 2020/03/20 · arxiv updated 2020/03/23 · openalex created_date 2020/03/27 · openalex updated_date 2026/07/28

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 γ_n,k(p,q) by (p+q)^k. As a follow-up, in 2012 Shin and Zeng showed that the fraction γ_n,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.

Citations

• Gamma-positivity in combinatorics and geometry

Cited by

• Counting permutations by alternating runs via Hetyei-Reiner trees

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