On the excedance sets of colored permutations

TL;DR

This paper generalizes the concept of excedance sets to colored permutations, providing formulas for counting permutations with specific excedance patterns and exploring their symmetric properties like log-concavity and unimodality.

Contribution

It introduces the excedance set and word for colored permutations and derives enumeration formulas using inclusion-exclusion, also analyzing their symmetric properties.

Findings

Derived formulas for counting colored permutations with given excedance words.

Proved log-concavity and unimodality of certain excedance word sequences.

Extended previous work on excedance sets to colored permutation groups.

Abstract

We define the excedence set and the excedance word on $G_{r,n}$, generalizing a work of Ehrenborg and Steingrimsson and use the inclusion-exclusion principle to calculate the number of colored permutations having a prescribed excedance word. We show some symmetric properties as Log concavity and unimodality of a specific sequence of excedance words.