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

TL;DR

This work analyzes the descent statistic $des(\pi)$ within conjugacy classes of the symmetric group $S_n$. It first establishes that the minimum $des$ value in any nontrivial class is $1$ via a constructive Young-diagram filling tied to the cycle structure, and it connects to Lyndon words through the explicit count $L_n=\frac{1}{n}\sum_{d|n}\mu(n/d)2^{d}$. It then proves a continuity property: for every nontrivial conjugacy class, the set of attainable $des$-values forms a complete interval from $1$ up to the class maximum $M$, using a bound $|des(\pi)-des(s_i^{-1}\pi s_i)|\le 2$ and a cyclic-descent framework to close gaps. The main result is a constructive path, combining Coxeter-conjugations and cyclic shifts, showing all intermediate descent values can be realized within each class, with implications for the combinatorial structure of conjugacy classes in $S_n$.

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.

Theorems & Definitions (30)

• Theorem 2.1
• Definition 2.2
• proof : proof of theorem \ref{['1des']}
• Example 2.3
• Definition 3.1
• Proposition 3.2
• Example 3.3
• proof : proof of proposition \ref{['d=012']}
• Theorem 3.4
• Proposition 3.5