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arXiv · 2310.04969

The Binomial-Stirling-Eulerian Polynomials

Kathy Q. Ji·Zhicong Lin
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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.

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Kathy Q. Ji, Zhicong Lin. 2023-10-08. The Binomial-Stirling-Eulerian Polynomials. https://arxiv.org/abs/2310.04969

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