Number of elements of order $r$ in the symmetric group $S_n$

Question

The question I am asking is already asked here.

I want specially the general part, i.e. number of elements of order $r$ in the symmetric group $S_n$, $n\ge 4$. Is there a general rule? Thanks in advance.

Answer 1

Every subset $S\subseteq\{1,\ldots,n\}$ with $\operatorname{lcm} S=n$ and map $S\to\Bbb N$ with $\sum_s sf(s)=n$ contributes $\frac{n!}{\prod_s s!^{f(s)}f(s)!}$ to the total count ...