Using recurrence relations to count in symmetric groups

We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group $S_n$ in order to construct recurrence relations for enumerating certain subsets of $S_n$. Occasionally one can find `closed form' solutions to such recurrence relations. For example, the probability that a random element of $S_n$ has no cycle of length divisible by $q$ is $\prod_{d=1}^{\lfloor n/q\rfloor} (1-\frac{1}{dq})$.