Width-k Eulerian polynomials of type A and B and its Gamma-positivity

TL;DR

This paper introduces generalized Eulerian polynomials of types A, B, and D based on new width-k descent and inversion statistics, proving their Gamma-positivity and providing combinatorial interpretations via quasisymmetric functions.

Contribution

It presents novel width-k Eulerian polynomials of types A, B, and D, establishing their Gamma-positivity and linking them to quasisymmetric functions and P-partitions.

Findings

Proved Gamma-positivity of width-k Eulerian polynomials.

Derived new generalizations of classical Eulerian polynomials.

Provided combinatorial interpretations using quasisymmetric functions.

Abstract

We define some generalizations of the classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [20] and called after width-k descents and width-k inversionsof type A in Davis's work [8]. Using the aforementioned new statistics, we derive some new generalizations of Eulerian polynomials of type A, B and D. It should also be noticed that we establish the Gamma-positivity of the "width-k" Eulerian polynomials and we give a combinatorial interpretation of finite sequences associated to these new polynomials using quasisymmetric functions and P-partition in Petersen's work [18].