Number of Elements with given Cycle Type

Theorem
Let $n \in \N$ be a natural number.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\lambda$ be an integer partition of $n$ such that $\lambda$ has $a_j$ parts of size $j$ for each $j$.

That is, such that there are $a_1$ instances of $1$s, $a_2$ instances of $2$s, $a_3$ instances of $3$s, and so on.

Let $C$ be the conjugacy class in $S_n$ comprising the elements whose cycle type is $\lambda$.

Then:
 * $\size C = \dfrac {n!} {\displaystyle \prod_j j^{a_j} a_j!}$