On γ-positive polynomials arising in Pattern Avoidance

A classical result of Foata and Schützenberger states that the γ-coefficients of the Eulerian polynomials enumerate permutations without double descents by the number of descents. In this paper, based on works of Cheng et al. and Stankova, we provide similar combinatorial interpretations for the γ  -coefficients of the inversion polynomials on 321-avoiding permutations and the descent polynomials on Separable permutations (or (2413,3142)(2413,3142)-avoiding permutations) and (1342,2431)(1342,2431)-avoiding permutations. Some further open problems are also proposed.