Some -positive polynomials arising from enumerations of the pseudo Schr?der paths

We introduce a notion of pseudo Schroder paths, which are counted by the Schroder numbers. Using the symbolic method and the Lagrange inversion formula, we provide an alternative proof that the descent polynomials of separable permutations are gamma-positive. We also present several classes of gamma-positive polynomials via enumerating the pseudo Schroder paths according to several combinatorial statistics by applying Riordan arrays.(c) 2023 Elsevier B.V. All rights reserved.