Prove that the probability that two elements of the symmetric group $\displaystyle S_n,$ chosen randomly with replacement, commute is $\displaystyle \frac{P(n)}{n!},$ where $\displaystyle P(n)$ is the partition function.
Prove that the probability that two elements of the symmetric group $\displaystyle S_n,$ chosen randomly with replacement, commute is $\displaystyle \frac{P(n)}{n!},$ where $\displaystyle P(n)$ is the partition function.
$\displaystyle P(n) $ is the number of cycle types of $\displaystyle S_n $. And the number of cycle types is the number of ways of choosing two elements of $\displaystyle S_n $ that commute.