= 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 $\{1,\ldots,N\}$. For length $\ell$, choose the other $\ell-1$ objects, arrange their order around the tagged object, and permute all remaining objects. The count is $\binom{N-1}{\ell-1}(\ell-1)!(N-\ell)!=(N-1)!$, independent of $\ell$. Its mean is $(N+1)/2$, while the mean number of other elements in that cycle is $(N-1)/2$.
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