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 (des,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.