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.

Read more about this topic: Random Permutation Statistics

Famous quotes containing the words number of, number and/or permutations:

“Strange goings on! Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight. What he did was butter a piece of toast. We are too familiar with the language of action to notice at first an anomaly: the ‘it’ of ‘Jones did it slowly, deliberately,...’ seems to refer to some entity, presumably an action, that is then characterized in a number of ways.”
—Donald Davidson (b. 1917)

“I believe if we introduced the Lord’s Prayer here, senators would propose a large number of amendments to it.”
—Henry Wilson (1812–1875)

“Motherhood in all its guises and permutations is more art than science.”
—Melinda M. Marshall (20th century)