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arXiv · 1812.09098

Around the $q$-binomial-Eulerian polynomials

Zhicong Lin·David G. L. Wang·Jiang Zeng
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Abstract

We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$\gamma$-positivity using group actions. Motivated by the sign-balance identity of D\'esarm\'enien--Foata--Loday for the $(\mathrm{des}, \mathrm{inv})$-Eulerian polynomials, we further investigate the sign-balance of the $q$-binomial-Eulerian polynomials. We show the unimodality of the resulting signed binomial-Eulerian polynomials by exploiting their continued fraction expansion and making use of a new quadratic recursion for the $q$-binomial-Eulerian polynomials. We finally use the method of continued fractions to derive a new $(p,q)$-extension of the $\gamma$-positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations.

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Zhicong Lin, David G. L. Wang, Jiang Zeng. 2018-12-21. Around the $q$-binomial-Eulerian polynomials. https://arxiv.org/abs/1812.09098

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