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!



The distribution of cycle length in random derangement
The number of permutations where $x$ is in a cycle of length $k$ is $(N-1)!$. In order to have a derangement, $k$ must be at least 2 and the remaining $N-k$ elements must be "deranged". When $N-k$ is not too small, the fraction of derangements will be very close to $1/e$, so the cycle length in a derangement will be asymptotically uniformly distributed in the sense that if we divide by $N$, it converges in distribution to uniform on $[0,1]$. But for $k$ close to $N$, there will be irregularities, for instance the cycle length is never $N-1$. 


