The Binomial-Stirling-Eulerian Polynomials

2023/10/08 by Kathy Q. Ji, Ji, Kathy Q., Lin, Zhicong · 1 citation
Agricultural and Biological Sciences · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2310.04969

openalex publication_date 2023/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the binomial-Stirling-Eulerian polynomials, denoted A_n(x,y|α), which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When α=1, these polynomials reduce to the binomial-Eulerian polynomials A_n(x,y), originally named by Shareshian and Wachs and explored by Chung-Graham-Knuth and Postnikov-Reiner-Williams. We investigate the γ-positivity of A_n(x,y|α) from two aspects: firstly by employing the grammatical calculus introduced by Chen; and secondly by constructing a new group action on permutations. These results extend the symmetric Eulerian identity found by Chung, Graham and Knuth, and the γ-positivity of A_n(x,y) first demonstrated by Postnikov, Reiner and Williams.

Cited by

• Symmetric decompositions and Euler-Stirling statistics on Stirling permutations

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