Inversions in Colored Permutations, Derangements, and Involutions

TL;DR

This paper extends classical permutation statistics to colored permutations, deriving formulas and generating functions for inversions in derangements and involutions within a unified combinatorial framework.

Contribution

It introduces a bijective approach to analyze inversion distributions in colored permutations, including derangements and involutions, extending classical results to the colored case.

Findings

Derived explicit formulas for inversion counts in colored derangements and involutions.

Established recurrence relations and generating functions for colored Mahonian numbers.

Unified combinatorial framework for analyzing permutation statistics in colored groups.

Abstract

Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups. In this work, we study the distribution of this statistic over colored permutations, including derangements and involutions. By establishing a bijective correspondence between colored permutations and colored Lehmer codes, we develop a unified framework for enumerating colored Mahonian numbers and analyzing their combinatorial properties. We derive explicit formulas, recurrence relations, and generating functions for the number of inversions in these families, extending classical results to the colored setting. We conclude with explicit expressions for inversions in colored derangements and involutions.