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- W4287026173 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 $A(n)=Acap [n]$, the limit $rho=lim_{ntoinfty}mid A(n)mid/n$ (if it esists) is usually called the density of set $A$. (Here $mid Bmid$ stands for the cardinality of the set $B$.) Several studies show that the asymptotic behavior of the cardinality $mid S_{n,A}mid$, as $ntoinfty$, 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 $sum_{kle n}mid S_{k,mathcal{P}}mid/k!$ as $ntoinfty$. In our proof we employ the classical Hardy-Littlewood-Karamata Tauberian theorem." @default.
- W4287026173 created "2022-07-25" @default.
- W4287026173 creator A5012982414 @default.
- W4287026173 date "2021-08-11" @default.
- W4287026173 modified "2023-09-29" @default.
- W4287026173 title "A Note on the Number of Permutations whose Cycle Lengths Are Prime Numbers" @default.
- W4287026173 doi "https://doi.org/10.48550/arxiv.2108.05291" @default.
- W4287026173 hasPublicationYear "2021" @default.
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