Inversions in Colored Permutations, Derangements, and Involutions

TL;DR

This work extends classical inversion statistics to colored permutations by developing a unified framework via colored Lehmer codes for the wreath product ${C_c\wr \mathfrak{S}_n}$. It establishes exact generating functions, recurrences, and closed forms for the distribution of the colored inversion statistic ${\rm inv}_c$, including detailed analyses for derangements and involutions. Key contributions include explicit formulas for the total number of colored inversions ${I_{c,n}}$, the total inversions in colored derangements ${t_n^{(c)}}$, and the total inversions in colored involutions ${i_n^{(c)}}$, together with their generating functions and symmetry properties. The results generalize MacMahon-type identities to arbitrary colorings, providing combinatorial and algebraic tools for enumerating colored Mahonian numbers and related statistics, with potential applications in symmetric group generalizations and related combinatorial structures.

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.

Figures (1)

• Figure 1: Reconstructing the permutation from its Lehmer code.

Theorems & Definitions (32)

• Definition 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Theorem 3.1: Symmetry
• proof
• Theorem 3.2: Summation property
• proof
• Theorem 3.3: Recurrence relation
• proof