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

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.