Random Permutation Statistics - Number of Permutations That Are Involutions

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 ∈ S_n:

$I(n) = n! g(z) = n! \sum_{a+2b=n} \frac{1}{a! \; 2^b \; b!} = n! \sum_{b=0}^{\lfloor n/2 \rfloor} \frac{1}{(n-2b)! \; 2^b \; b!}.$

Dividing by n! yields the probability that a random permutation is an involution.

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