Commuting Eulerian operators

Abstract

Motivated by the work of Visontai and Dey-Sivasubramanian on the gamma-positivity of some polynomials, we find the commutative property of a pair of Eulerian operators. As an application, we show the bi-gamma-positivity of the descent polynomials on permutations of the multiset $\{1^{a_1},2^{a_2},\dots ,n^{a_n}\}$, where $0⩽a_i⩽2$. Therefore, these descent polynomials are all alternatingly increasing, and so they are unimodal with modes in the middle.

9 pages