  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
$\prod_{d=1}^{\lfloor n/q\rfloor} (1-\frac{1}{dq})$.
