Random Permutation Statistics - Expected Cycle Size of A Random Element

Expected Cycle Size of A Random Element

We select a random element q of a random permutation and ask about the expected size of the cycle that contains q. Here the function is equal to, because a cycle of length k contributes k elements that are on cycles of length k. Note that unlike the previous computations, we need to average out this parameter after we extract it from the generating function (divide by n). We have

$\frac{\partial}{\partial u} g(z, u) \Bigg|_{u=1} = \frac{1}{1-z} \sum_{k\ge 1} k^2 \frac{z^k}{k} = \frac{1}{1-z} \frac{z}{(1-z)^2} = \frac{z}{(1-z)^3}.$

Hence the expected length of the cycle that contains q is

$\frac{1}{n} \frac{z}{(1-z)^3} = \frac{1}{n} \frac{1}{2} n (n+1) = \frac{1}{2} (n+1).$

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