Expected Number of Cycles of A Given Size m
In this problem we use a bivariate generating function g(z, u) as described in the introduction. The value of b for a cycle not of size m is zero, and one for a cycle of size m. We have
or
This means that the expected number of cycles of size m in a permutation of length n less than m is zero (obviously). A random permutation of length at least m contains on average 1/m cycles of length m. In particular, a random permutation contains about one fixed point.
The OGF of the expected number of cycles of length less than or equal to m is therefore
where Hm is the mth harmonic number. Hence the expected number of cycles of length at most m in a random permutation is about ln m.
Read more about this topic: Random Permutation Statistics
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