The $\gamma$-positivity of basic Eulerian polynomials via group actions

TL;DR

This paper offers a combinatorial interpretation of the $oldsymbol{eta}$-coefficients of basic Eulerian polynomials, refining classical results by employing permutation group actions and new triple statistics.

Contribution

It introduces a combinatorial framework for understanding $oldsymbol{eta}$-coefficients of Eulerian and derangement polynomials using group actions and triple statistics.

Findings

Provides combinatorial interpretation for $oldsymbol{eta}$-coefficients.

Refines classical $oldsymbol{eta}$-positivity results.

Utilizes Brändén's modified Foata–Strehl action and triple statistics.

Abstract

We provide combinatorial interpretation for the $\gamma$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $\gamma$-coefficients for derangements. Our results refine the classical $\gamma$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Br\"and\'en's modified Foata--Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).