Arithmetic functions and fixed points of powers of permutations

TL;DR

This paper explores how the fixed points of permutation powers uniquely determine the permutation's conjugacy class, provides an algorithm for this reconstruction, and characterizes which arithmetic functions can arise as fixed point counts.

Contribution

It introduces a method to identify permutation conjugacy classes from fixed point functions and characterizes fixed point counting functions of permutations.

Findings

The fixed point count function determines the conjugacy class of a permutation.

An algorithm is constructed to compute the conjugacy class from fixed point data.

The paper characterizes all arithmetic functions that can be fixed point counts of permutations.

Abstract

Let $\sigma$ be a permutation of a nonempty finite or countably infinite set $X$ and let $F_X\left( \sigma^k\right)$ count the number of fixed points of the $k$th power of $\sigma$. This paper explains how the arithmetic function $k \mapsto \left(F_X\left( \sigma^k\right) \right)_{k=1}^{\infty}$ determines the conjugacy class of the permutation $\sigma$, constructs an algorithm to compute the conjugacy class from the fixed point counting function $F_X\left( \sigma^k\right)$, and describes the arithmetic functions that are fixed point counting functions of permutations.