## Compute the number of conjugacy classes of elements of prime order in Sym(n)

Let $p$ be a prime. Find a formula for the number of conjugacy classes of elements of order $p$ in $S_n$ using the floor function.

Every element of order $p$ in $S_n$ is a product of commuting $p$-cycles. Provided $mp \leq n$, there is a conjugacy class in $S_n$ consisting of all products of $m$ commuting $p$-cycles; that is, one class for all integers $1\leq m \leq n/p$. Thus the number of such classes is $\lfloor n/p \rfloor$.