(Am not mathematician, sorry in advance for the sloppy notations.)

Consider the class of permutations of $n$ elements such that for each permutation $\pi$ in this class, and for each $x$ in $\{1,\dots,n\}$, $\pi(x)$ can take only two fixed distinct values.

My question is: what's the expected number of cycles of size $m$ for such a class of permutations? In particular, does it differ from the case of a uniformly random permutation?

Guess that a first step is to determine the size of these classes. For $n=2^k$, one can show that such a class contains $2^{2^{n-1}}$ such permutations (see each element as a $k$-bit string, and the permutation as a nonsingular feedback shift register, then count the number of distinct feedback functions).

Thanks for any hint or reference to related previous work.