Some γ-positive polynomials arising from enumerations of the pseudo Schröder paths

We introduce a notion of pseudo Schröder paths, which are counted by the Schröder numbers. Using the symbolic method and the Lagrange inversion formula, we provide an alternative proof that the descent polynomials of separable permutations are γ-positive. We also present several classes of γ-positive polynomials via enumerating the pseudo Schröder paths according to several combinatorial statistics by applying Riordan arrays.