Around the $q$-binomial-Eulerian polynomials

We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$γ$-positivity using group actions. Motivated by the sign-balance identity of Désarménien--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 $γ$-positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations.

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