Commuting Eulerian operators

Motivated by the work of Visontai and Dey-Sivasubramanian on the gamma-positivity of some polynomials, we discover 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 {1a1 , 2a2, ... , nan }, where 0 5 ai 5 2. Therefore, these descent polynomials are all alternatingly increasing, and so they are unimodal with modes in the middle. (c) 2023 Elsevier B.V. All rights reserved.