Cyclotomic Euler-Mahonian polynomials

TL;DR

This paper introduces formulas for cyclotomic Euler-Mahonian polynomials, extending previous results to include both even and odd cases, and provides new signed polynomial formulas using Hadamard products.

Contribution

It derives a new formula for cyclotomic Euler-Mahonian polynomials based on Hadamard products, including signed versions for both even and odd cases.

Findings

Derived a Hadamard product-based formula for cyclotomic Euler-Mahonian polynomials.

Extended Wachs' formula to include the odd case for signed Euler-Mahonian polynomials.

Obtained the $I$-analogue of Wachs' formula for signed polynomials.

Abstract

The cyclotomic Eulerian polynomials and the cyclotomic Mahonian polynomials have each been the subject of extensive studies in Combinatorics, with particular attention to their signed versions. In contrast, the joint study of cyclotomic Euler-Mahonian polynomials has received far less consideration. To the best of our knowledge, the only prior result in this direction is a formula due to Wachs for the signed Euler-Mahonian polynomials in the even case. In this paper, we focus on the cyclotomic Euler-Mahonian polynomials and derive a formula based on the Hadamard product. As corollaries, we obtain the $I$-analogue (where $I=\sqrt{-1}$) of Wachs' formula for signed Euler-Mahonian polynomials, as well as the previously missing odd case for the signed Euler-Mahonian polynomials.