The Binomial-Stirling-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.

arXiv comment18 page. Any comments are welcome. arXiv admin note: text overlap with arXiv:2310.01053