The characters of symmetric groups that depend only on length

TL;DR

This paper characterizes the characters of symmetric groups that depend solely on the number of cycles in a permutation, providing a complete description of such characters.

Contribution

It offers a complete determination of symmetric group characters that depend only on cycle count, a specific class of characters.

Findings

Characters depend only on cycle count are fully characterized.

Provides explicit formulas for these characters.

Enhances understanding of symmetric group representations.

Abstract

The characters $\chi(\pi)$ of $S_n$ that depend only on the number of cycles of $\pi$ are completely determined.