The Binomial-Stirling-Eulerian Polynomials

TL;DR

This paper introduces a new family of polynomials called binomial-Stirling-Eulerian polynomials, explores their properties, and proves their gamma-positivity using grammatical calculus and permutation group actions.

Contribution

It defines the binomial-Stirling-Eulerian polynomials and establishes their gamma-positivity through novel combinatorial and algebraic methods.

Findings

Extension of symmetric Eulerian identities

Proof of gamma-positivity for the new polynomials

Connection to existing binomial-Eulerian polynomials

Abstract

We introduce the binomial-Stirling-Eulerian polynomials, denoted $\tilde{A}_n(x,y|{\alpha})$, which encompass binomial coefficients, Eulerian numbers and two Stirling statistics: the left-to-right minima and the right-to-left minima. When $\alpha=1$, these polynomials reduce to the binomial-Eulerian polynomials $\tilde{A}_n(x,y)$, originally named by Shareshian and Wachs and explored by Chung-Graham-Knuth and Postnikov-Reiner-Williams. We investigate the $\gamma$-positivity of $\tilde{A}_n(x,y|{\alpha})$ 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 $\gamma$-positivity of $\tilde{A}_n(x,y)$ first demonstrated by Postnikov, Reiner and Williams.