Eulerian polynomials for multipermutations and signed multipermutations

TL;DR

This paper investigates Eulerian polynomials associated with permutations and signed permutations of a multiset, establishing their properties such as recurrence relations and unimodality with a unified proof.

Contribution

It provides a comprehensive analysis of Eulerian polynomials for multipermutations and signed multipermutations, including new proofs of unimodality.

Findings

Proved recurrence relations for these polynomials

Established unimodality with modes in the middle

Unified proof of key polynomial properties

Abstract

In this paper, we study Eulerian polynomials for permutations and signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$. Properties of these polynomials, including recurrence relations and unimodality are discussed. In particular, we give a unified proof of the fact that these polynomials are unimodal with modes in the middle.