 Cycle length of a tagged element in a uniform permutation

In a uniformly random permutation of $N$ objects, the cycle containing one specified object has a uniform length on $\{1,\dots ,N\}$. For length $\ell$, choose the other $\ell -1$ objects, arrange their order around the tagged object, and permute all remaining objects. The count is $\left(\genfrac{}{}{0ex}{}{N-1}{\ell -1}\right)(\ell -1)!(N-\ell )!=(N-1)!$, independent of $\ell$. Its mean is $(N+1)/2$, while the mean number of other elements in that cycle is $(N-1)/2$.