Moments of Fixed Points
The mixed GF of the set of permutations by the number of fixed points is
Let the random variable X be the number of fixed points of a random permutation. Using Stirling numbers of the second kind, we have the following formula for the mth moment of X:
where is a falling factorial. Using, we have
which is zero when, and one otherwise. Hence only terms with contribute to the sum. This yields
Read more about this topic: Random Permutation Statistics
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