This isn't an answer to your question, but without any restrictions, the expected number of $r$-cycles in a permutation is $\frac{1}{r}$, so the expected number of cycles in a permutation in $S_n$ is $1 + \frac{1}{2} + ... + \frac{1}{n} = H_n \sim \log n$.  So this provides a heuristic upper bound on the number you're actually looking for.