It is known that for a fixed x $\in \{0,1,...,N-1\}$, the length of the cycle of x in a random permutation in $S_N$ distributes uniformly in $\{1, . . . ,N\}$.

My question is regarding the length of x in a random ** derangement** (permutation without any fixed point).

Does the length distributes uniformly in $\{2, . . . ,N\}$? If not - what is the distribution?

Any proof, proof sketch, reference or good explanation will be appreciated. I tried to google it, or to find relevant papers in google-scholar, but without success.

Thanks in advance!