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