# Math Help - Probabilistic Group Theory

1. ## Probabilistic Group Theory

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

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