I'm interested in the number of permutations for a specified number of fixed points and cycles.
Suppose we are in $S_n$. For any permutation in $S_n$, let $h$ be the number of changed points (the complement of fixed points) and $N$ be the number of cycles of the permutation with length no less than 2 (number of cycles for the changed points). Therefore, for a given $h$, the number of cycles $N$ can range from $1$ to $\lfloor h/2\rfloor$.
So my question is how many permutations are there in $S_n$ for a given $h$ and $N$? If the exact expression is complicated, can we derive a concise and sharp upper bound of the number in terms of $n,h,N$?