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:

    “I who have been involved with all styles of painting can assure you that the only things that fluctuate are the waves of fashion which carry the snobs and speculators; the number of true connoisseurs remains more or less the same.”
    —Pablo Picasso (1881–1973)

    “Even in ordinary speech we call a person unreasonable whose outlook is narrow, who is conscious of one thing only at a time, and who is consequently the prey of his own caprice, whilst we describe a person as reasonable whose outlook is comprehensive, who is capable of looking at more than one side of a question and of grasping a number of details as parts of a whole.”
    —G. Dawes Hicks (1862–1941)

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