The asymptotic of the number of permutations whose cycle lengths are prime numbers

Abstract

Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. Furthermore, let $\mid A(n)\mid$ denote the cardinality of the set $A(n)=A\cap [n]$. The limit $\rho=\lim_{n\to\infty}\mid A(n)\mid/n$ (if it exists) is called the density of set $A$. It turns out that, as $n\to\infty$, the cardinality $\mid S_{n,A}\mid$ of the set $S_{n,A}$ essentially depends on $\rho$. The case $\rho>0$ was studied by several authors under certain additional conditions on $A$. In 1999, Kolchin noticed that there is a lack studies on classes of permutations for which $\rho=0$. In this context, he also proposed investigations on certain particular cases. In this paper, we consider the permutations whose cycle lengths are prime numbers, that is, we assume that $A=\mathcal{P}$, where $\mathcal{P}$ denotes the set of all primes. For this class of permutations, the Prime Number Theorem implies that $\rho=0$. In this paper, we show that, as $n\to\infty$, the ratio $S_{n,\mathcal{P}}/(n-1)!$ approaches a finite limit and determine its value explicitly. Our method of proof employs classical Tauberian theorems.