Given a positive integer $n$, let $P(n)$ be the number of partitions of $n$. I am curious as to why that, for any given positive integer $n$, the number of distinct conjugacy classes for elements of $S_n$ is exactly $P(n)$?
I can see this is true for $n=5$. Right, $P(5) = 7$, and there are exactly $7$ conjugacy classes of $S_5$. I guess I'm curious as to why this generalizes for any positive integer $n$.