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 ∈ Sn:

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:

    “Hence, a generative grammar must be a system of rules that can iterate to generate an indefinitely large number of structures. This system of rules can be analyzed into the three major components of a generative grammar: the syntactic, phonological, and semantic components.”
    —Noam Chomsky (b. 1928)

    “Take away from the courts, if it could be taken away, the power to issue injunctions in labor disputes, and it would create a privileged class among the laborers and save the lawless among their number from a most needful remedy available to all men for the protection of their business interests against unlawful invasion.... The secondary boycott is an instrument of tyranny, and ought not to be made legitimate.”
    —William Howard Taft (1857–1930)

    “The new shopping malls make possible the synthesis of all consumer activities, not least of which are shopping, flirting with objects, idle wandering, and all the permutations of these.”
    —Jean Baudrillard (b. 1929)