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:

    “The basis of successful relief in national distress is to mobilize and organize the infinite number of agencies of self help in the community. That has been the American way.”
    —Herbert Hoover (1874–1964)

    “At thirty years a woman asks her lover to give her back the esteem she has forfeited for his sake; she lives only for him, her thoughts are full of his future, he must have a great career, she bids him make it glorious; she can obey, entreat, command, humble herself, or rise in pride; times without number she brings comfort when a young girl can only make moan.”
    —Honoré De Balzac (1799–1850)

    “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)