Number of Permutations That Are Involutions
An involution is a permutation σ so that σ2 = 1 under permutation composition. It follows that σ may only contain cycles of length one or two, i.e. the EGF g(z) of these permutations is
This gives the explicit formula for the total number of involutions among the permutations σ ∈ Sn:
Dividing by n! yields the probability that a random permutation is an involution.
Read more about this topic: Random Permutation Statistics
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