Colored Multiset Eulerian Polynomials

TL;DR

This paper introduces and characterizes colored multiset Eulerian polynomials, establishing their distributional properties and conditions for self-interlacing, with applications to $s$-Eulerian polynomials and $eta$-coefficients.

Contribution

It generalizes existing Eulerian polynomials, characterizes symmetric cases, and provides new conditions for self-interlacing and related properties.

Findings

Colored multiset Eulerian polynomials satisfy real-rootedness, log-concavity, and unimodality.

Symmetric colored multiset Eulerian polynomials are characterized and linked to self-interlacing.

Applications include generalized answers to questions on $s$-Eulerian polynomials and $eta$-coefficients.

Abstract

Colored multiset Eulerian polynomials are a common generalization of MacMahon's multiset Eulerian polynomials and the colored Eulerian polynomials, both of which are known to satisfy well-studied distributional properties including real-rootedness, log-concavity and unimodality. The symmetric colored multiset Eulerian polynomials are characterized and used to prove sufficient conditions for a colored multiset Eulerian polynomial to be self-interlacing. The latter property implies the aforementioned distributional properties as well as others, including the alternatingly increasing property and bi-$\gamma$-positivity. To derive these results, multivariate generalizations of an identity due to MacMahon are deduced. The results are applied to a pair of questions, both previously studied in several special cases, that are seen to admit more general answers when framed in the context of colored multiset Eulerian polynomials. The first question pertains to $s$-Eulerian polynomials, and the second to interpretations of $\gamma$-coefficients.