Signed Mahonian Polynomials on Colored Derangements

Abstract

The polynomial ${\sum }_{\ddot{I}€\in W}{q}^{maj(\ddot{I}€)}$ 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 ${\sum }_{\ddot{I}€\in W}(-1{)}^{l(\ddot{I}€)}{q}^{maj(\ddot{I}€)}$ 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 $$\sum _{\ddot{I}€\in D_n^{(c)}}(-1{)}^{L(\ddot{I}€)}{q}^{fmaj(\ddot{I}€)}$$ 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.