Width-$k$ Eulerian polynomials of type $A$ and $B$: The $\gamma$-positivity

Source abstract

In this paper, we introduce some new generalizations of classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [18], and called $k$-width descents and $k$-width inversions of type $A$ [8]. Using the aforementioned new statistics, we derive new generalizations of Eulerian polynomials of type $A$, $B$ and $D$. We establish also the $\gamma$-positivity of the Eulerian "width-$k$" polynomials. Referring to Petersen's paper [16], we give a combinatorial interpretation of finite sequences associated with these new polynomials using quasi-symmetric functions and a partition $P$. Received: 17 October 2022 | Accepted: 21 March 2023