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:

    Accidents will occur in the best-regulated families; and in families not regulated by that pervading influence which sanctifies while it enhances ... in short, by the influence of Woman, in the lofty character of Wife, they may be expected with confidence, and must be borne with philosophy.
    Charles Dickens (1812–1870)

    No Government can be long secure without a formidable Opposition. It reduces their supporters to that tractable number which can be managed by the joint influences of fruition and hope. It offers vengeance to the discontented, and distinction to the ambitious; and employs the energies of aspiring spirits, who otherwise may prove traitors in a division or assassins in a debate.
    Benjamin Disraeli (1804–1881)

    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)

    When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.
    Henry David Thoreau (1817–1862)

    And catch the gleaming of a random light,
    That tells me that the ship I seek is passing, passing.
    Paul Laurence Dunbar (1872–1906)