Colored Multiset Eulerian Polynomials
Danai Deligeorgaki, Bin Han, and Liam Solus

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--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…
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Taxonomy
TopicsAdvanced Mathematical Identities
