论文标题
循环长度为质数的排列数量的渐近数
The asymptotic of the number of permutations whose cycle lengths are prime numbers
论文作者
论文摘要
令$ a $为一组自然数字，让$ s_ {n，a} $是$ [n] = \ {1,2，...，n \} $的所有排列的集合，属于$ a $。此外，令$ \ mid a（n）\ mid $表示集合$ a（n）= a \ cap [n] $的基数。限制$ρ= \ lim_ {n \ to \ infty} \ mid a（n）\ mid/n $（如果存在）称为集合$ a $的密度。事实证明，作为$ n \ to \ infty $，基数$ \ mid s_ {n，a} \ mid $的$ s_ {n，a} $基本上取决于$ρ$。 $ρ> 0 $在某些其他条件下对$ a $进行了研究。 1999年，科尔钦（Kolchin）注意到，缺乏对$ρ= 0 $的排列类别的研究。在这种情况下，他还建议对某些特定情况进行调查。在本文中，我们考虑其循环长度为质量数字的排列，也就是说，我们假设$ a = \ Mathcal {p} $，其中$ \ Mathcal {p} $表示所有素数的集合。对于此类排列，质数定理意味着$ρ= 0 $。在本文中，我们表明，作为$ n \ to \ infty $，比率$ s_ {n，\ mathcal {p}}/（n-1）！$接近有限限制并明确确定其值。我们的证明方法采用古典陶伯里亚定理。
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 $ρ=\lim_{n\to\infty}\mid A(n)\mid/n$ (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 $ρ$. The case $ρ>0$ 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 $ρ=0$. 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 $ρ=0$. 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.
