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arXiv · 1702.06666

Gamma-positivity of variations of Eulerian polynomials

John Shareshian·Michelle L. Wachs
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Abstract

An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Schützenberger shows that the Eulerian polynomials have a stronger property, namely $γ$-positivity, and a formula of Postnikov, Reiner and Williams does the same for the binomial-Eulerian polynomials. We obtain $q$-analogs of both the Foata-Schützenberger formula and an alternative to the Postnikov-Reiner-Williams formula, and we show that these $q$-analogs are specializations of analogous symmetric function identities. Algebro-geometric interpretations of these symmetric function analogs are presented.

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John Shareshian, Michelle L. Wachs. 2018-06-12. Gamma-positivity of variations of Eulerian polynomials. https://arxiv.org/abs/1702.06666

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