Signed Mahonian Polynomials on Colored Derangements
Hasan Arslan, Nazmiye Alemdar
TL;DR
The paper extends Mahonian-type $q$-enumeration to colored derangements in the group $G_{c,n}$ via the flag-major index $fmaj$ and a length function $L$. It derives an explicit signed Mahonian formula when $c$ is even, expressed as a product of $q$-integers times a Gauss-inversion-derived sum, and uses this to obtain counts for even/odd derangements and their difference, with connections to known results for Weyl groups. The work highlights parity effects, discusses the absence of a neat product formula for odd $c$, and provides supporting Appendix data that suggests symmetry properties between $fmaj$ and the inversion-like statistic in specific cases. Overall, it advances exact $q$-counting formulas for colored derangements and links them to classical combinatorial tools and Weyl-group theory.
Abstract
The polynomial $\sum_{π\in W}q^{maj(π)}$ 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 $\sum_{π\in W}(-1)^{l(π)}q^{maj(π)}$ 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 $$\sum_{π\in D_{n}^{(c)}}(-1)^{L(π)}q^{fmaj(π)}$$ 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 $G_{c,n}$ when $c$ is even.