Arithmetic functions and fixed points of powers 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 kth 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.