# Probabilistic Group Theory

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• November 11th 2009, 07:53 AM
NonCommAlg
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.
• November 15th 2009, 10:59 PM
Sampras
Quote:

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.