论文标题
列举副标置换
Enumerating coprime permutations
论文作者
论文摘要
如果$ \ gcd（m，σ（m））= 1 $ for $ m \ in [n] $，则将置换$σ$定义为coprime。在此注释中，证明了最新的猜想，我们证明了$ [n] $上的codrime排列数为$ n！ （p-2）^{（1-2/p）}}。
Define a permutation $σ$ to be coprime if $\gcd(m,σ(m)) = 1$ for $m\in[n]$. In this note, proving a recent conjecture of Pomerance, we prove that the number of coprime permutations on $[n]$ is $n!\cdot (c+o(1))^n$ where \[c = \prod_{p\text{ prime }}\frac{(p-1)^{2(1-1/p)}}{p\cdot (p-2)^{(1-2/p)}}.\] The techniques involve entropy maximization for the upper bound, and a mixture of number-theoretic bounds, permanent estimates, and the absorbing method for the lower bound.
