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.

arXiv commentThe main result of the paper (Theorem 3) is incorrect. This requires further studies on the underlying problem