Gamma-positivity and partial gamma-positivity of descent-type polynomials

TL;DR

This paper explores gamma-positivity in descent-type polynomials using grammatical methods, providing proofs, expansions, recurrences, and combinatorial interpretations for various classes of polynomials.

Contribution

It introduces a new grammatical approach to establish gamma-positivity and partial gamma-positivity for multiple polynomial families related to permutations and Stirling structures.

Findings

Proved gamma-positivity of Eulerian and type B Eulerian polynomials.

Derived partial gamma-positive expansions for Stirling permutation polynomials.

Provided recurrences and combinatorial interpretations for gamma-coefficients.

Abstract

In this paper, we study gamma-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present grammatical proofs of the gamma-positivity of the Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide partial gamma-positive expansions for several multivariate polynomials associated to Stirling permutations, Legendre-Stirling permutations, Jacobi-Stirling permutations and type B derangements, and the recurrences for the partial gamma-coefficients of these expansions are also obtained. Moreover, we define variants of the Foata-Strehl group action which are used to give combinatorial interpretations for the coefficients of most of these partial gamma-positive expansions.