The range of the des statistic for conjugacy classes in $S_n$

TL;DR

This paper characterizes the possible values of the descent statistic within each conjugacy class of the symmetric group, establishing the range, minimal, and maximal values, and providing a constructive method to realize all intermediate values.

Contribution

It determines the exact range of the descent statistic for all conjugacy classes in $S_n$ and introduces a constructive approach to attain each value.

Findings

Minimum descent statistic value is 1 for all non-identity classes.

Every integer between the minimum and maximum values is achievable.

The results have implications for the combinatorial structure of symmetric groups.

Abstract

We determine the range of the des statistic on every conjugacy class in the symmetric group $S_n$, prove that the minimum is $1$ (except for the identity class), and show that every intermediate value from $1$ to the maximum value is attained. We also demonstrate a constructive method to achieve every value in the range and discuss its combinatorial implications.