Signed Mahonian Polynomials on Colored Derangements

TL;DR

This paper studies signed Mahonian polynomials on colored derangements within colored permutation groups, deriving formulas for counting derangements of even length and the difference between even and odd length derangements.

Contribution

It introduces a new signed Mahonian polynomial on colored derangements and provides explicit formulas for counting even-length derangements and their length parity differences.

Findings

Derived a formula for counting colored derangements of even length when c is even.

Established a formula for the difference between even and odd length derangements for all positive c.

Connected Mahonian statistics with complex root systems in the context of colored permutation groups.

Abstract

The polynomial $\sum_{\pi \in W}q^{maj(\pi)}$ of major index over a classical Weyl group $W$ with a generating set $S$ is called the Mahonian polynomial over $W$, and also the polynomial $\sum_{\pi \in W}(-1)^{l(\pi)}q^{maj(\pi)}$ of major index together with sign over the group $W$ is called the signed Mahonian polynomial over the group $W$, where $l$ is the length function on $W$ defined in terms of the generating set $S$. We concern with the signed Mahonian polynomial $$\sum_{\pi \in D_{n}^{(c)}}(-1)^{L(\pi)}q^{fmaj(\pi)}$$ on the set $D_{n}^{(c)}$ of colored derangements in the group $G_{c,n}$ of colored permutations, where $L$ denotes the length function defined by means of a complex root system described by Bremke and Malle in $G_{c,n}$ and $fmaj$ defined by Adin and Roichman in $G_{c,n}$ represents the \textit{flag-major index}, which is a Mahonian statistic. As an application of the formula for signed Mahonian polynomials on the set of colored derangements, we will derive a formula to count colored derangements of even length in $G_{c,n}$ when $c$ is an even number. Finally, we conclude by providing a formula for the difference between the number of derangements of even and odd lengths in $G_{c,n}$ for every positive integer $c$, regardless of whether c is odd or even.