A Note on the Number of Permutations whose Cycle Lengths Are Prime Numbers

TL;DR

This paper investigates the asymptotic behavior of permutations with prime cycle lengths, deriving formulas for their counting function using advanced analytic methods, focusing on cases where the set of cycle lengths has zero density.

Contribution

It provides the first asymptotic analysis of permutations with prime cycle lengths, employing Tauberian theorems to handle the zero-density case.

Findings

Derived an asymptotic formula for the summatory function of permutations with prime cycle lengths.

Showed that the set of prime cycle lengths has zero density, affecting asymptotic behavior.

Applied Hardy-Littlewood-Karamata Tauberian theorem to obtain results.

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 $A$. For $A(n)=A\cap [n]$, the limit $\rho=\lim_{n\to\infty}\mid A(n)\mid/n$ (if it esists) is usually called the density of set $A$. (Here $\mid B\mid$ stands for the cardinality of the set $B$.) Several studies show that the asymptotic behavior of the cardinality $\mid S_{n,A}\mid$, as $n\to\infty$, depends on the density $\rho$. It turns out that the asumption $\rho>0$ plays an essential role in the asymptotic analysis of $\mid S_{n,A}\mid$. Kolchin (1999) noticed that there is a lack of studies on classes of permutations satisfying $\rho=0$ and proposed investigations on certain particular cases. In this note, 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. From the Prime Number Theorem it follows that $\rho=0$ for this class of permutations. We deduce an asymptotic formula for the summatory function $\sum_{k\le n}\mid S_{k,\mathcal{P}}\mid/k!$ as $n\to\infty$. In our proof we employ the classical Hardy-Littlewood-Karamata Tauberian theorem.