Random Permutation Statistics - Expected Number of Cycles of Any Length of A Random Permutation

Expected Number of Cycles of Any Length of A Random Permutation

We construct the bivariate generating function using, where is one for all cycles (every cycle contributes one to the total number of cycles).

Note that has the closed form

and generates the unsigned Stirling numbers of the first kind.

We have

\frac{\partial}{\partial u} g(z, u) \Bigg|_{u=1} = \frac{1}{1-z} \sum_{k\ge 1} b(k) \frac{z^k}{k} = \frac{1}{1-z} \sum_{k\ge 1} \frac{z^k}{k} = \frac{1}{1-z} \log \frac{1}{1-z}.

Hence the expected number of cycles is, or about .

Read more about this topic:  Random Permutation Statistics

Famous quotes containing the words expected, number, cycles, length and/or random:

    “The morning cup of coffee has an exhiliration about it which the cheering influence of the afternoon or evening cup of tea cannot be expected to reproduce.”
    —Oliver Wendell Holmes, Sr. (1809–1894)

    “Black lady,
    what will I do
    without your two flowers?
    I have inhabited you, number by number.
    I have pushed you in and out like a needle.”
    —Anne Sexton (1928–1974)

    “The stars which shone over Babylon and the stable in Bethlehem still shine as brightly over the Empire State Building and your front yard today. They perform their cycles with the same mathematical precision, and they will continue to affect each thing on earth, including man, as long as the earth exists.”
    —Linda Goodman (b. 1929)

    “At length to hospital
    This man was limited,
    Where screens leant on the wall
    And idle headphones hung.
    Since he would soon be dead
    They let his wife come along
    And pour out tea, each day.”
    —Philip Larkin (1922–1986)

    “It is a secret from nobody that the famous random event is most likely to arise from those parts of the world where the old adage “There is no alternative to victory” retains a high degree of plausibility.”
    —Hannah Arendt (1906–1975)