Labeled Partitions with Colored Permutations

TL;DR

This paper extends labeled partitions to colored permutations with cyclic structures, deriving generating functions for indices, exploring q-derangement relations, and providing combinatorial interpretations of known formulas.

Contribution

It introduces a new framework for analyzing colored permutations using labeled partitions, leading to novel generating functions and combinatorial insights.

Findings

Derived generating functions for maj_k indices of colored permutations

Established a combinatorial relation on q-derangement numbers for colored permutations

Provided a combinatorial interpretation of a formula by Adin, Gessel, and Roichman

Abstract

In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an involution on permutations that implies the generating function formula for the signed q-counting of the major indices due to Gessel and Simon. This involution can be extended to signed permutations. In this way, we obtain a combinatorial interpretation of a formula of Adin, Gessel and Roichman.