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

2021/08/11 by Ljuben Mutafchiev, Mutafchiev, Ljuben
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2108.05291

openalex publication_date 2021/08/11 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let A be a set of natural numbers and let S_n,A be the set of all\npermutations of [n]= 1,2,...,n with cycle lengths belonging to A. For\nA(n)=A\∩ [n], the limit \ρ=\lim_n\→\∞\| A(n)\|/n (if it\nesists) is usually called the density of set A. (Here \| B\| stands for\nthe cardinality of the set B.) Several studies show that the asymptotic\nbehavior of the cardinality \| S_n,A\|, as n\→\∞, depends on the\ndensity \ρ. It turns out that the asumption \ρ>0 plays an essential\nrole in the asymptotic analysis of \| S_n,A\|. Kolchin (1999) noticed\nthat there is a lack of studies on classes of permutations satisfying \ρ=0\nand proposed investigations on certain particular cases. In this note, we\nconsider the permutations whose cycle lengths are prime numbers, that is, we\nassume that A=\P, where \P denotes the set of all primes.\nFrom the Prime Number Theorem it follows that \ρ=0 for this class of\npermutations. We deduce an asymptotic formula for the summatory function\n\∑_{k\≤ n}\| S_k,\P\|/k! as n\→\∞. In our proof we\nemploy the classical Hardy-Littlewood-Karamata Tauberian theorem.\n

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