Commuting Eulerian operators

TL;DR

This paper explores the commutative properties of Eulerian operators and demonstrates the bi-gamma-positivity of certain descent polynomials, revealing their unimodal and alternatingly increasing nature.

Contribution

It establishes the commutative property of Eulerian operators and proves bi-gamma-positivity for descent polynomials on specific multiset permutations.

Findings

Descent polynomials are bi-gamma-positive.

These polynomials are alternatingly increasing.

They are unimodal with modes in the middle.

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 $\{1^{a_1},2^{a_2},\ldots,n^{a_n}\}$, where $0\leqslant a_i\leqslant 2$. Therefore, these descent polynomials are all alternatingly increasing, and so they are unimodal with modes in the middle.