# Around the $q$-binomial-Eulerian polynomials

## TL;DR

This paper provides a combinatorial interpretation and new proofs of properties of $q$-binomial-Eulerian polynomials, including $q$-$ ext{γ}$-positivity and unimodality, using group actions, continued fractions, and quadratic recursions.

## Contribution

It introduces a combinatorial interpretation, an alternative proof of $q$-$ ext{γ}$-positivity, and a new $(p,q)$-extension involving crossings and nestings of permutations.

## Key findings

- Established combinatorial interpretation of $q$-binomial-Eulerian polynomials.
- Proved $q$-$ ext{γ}$-positivity using group actions.
- Demonstrated unimodality via continued fraction expansion.

## 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.