γ-positivity and partial γ-positivity of descent-type polynomials

Abstract

In this paper, we study γ-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present a unified grammatical proof of the γ-positivity of Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide partial γ-positive expansions for several multivariate polynomials associated to Stirling permutations, Legendre-Stirling permutations, Jacobi-Stirling permutations and type B derangements. The recurrence relations for the partial γ-coefficients of these expansions are also obtained. By using some variants of the Foata-Strehl group action, we provide combinatorial interpretations for the coefficients of most of these partial γ-positive expansions.