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:

    Like sleep disturbances, some worries at separation can be expected in the second year. If you accept this, then you will avoid reacting to this anxiety as if it’s your fault. A mother who feels guilty will appear anxious to the child, as if to affirm the child’s anxiety. By contrast, a parent who understands that separation anxiety is normal is more likely to react in a way that soothes and reassures the child.
    Cathy Rindner Tempelsman (20th century)

    To make life more bearable and pleasant for everybody, choose the issues that are significant enough to fight over, and ignore or use distraction for those you can let slide that day. Picking your battles will eliminate a number of conflicts, and yet will still leave you feeling in control.
    Lawrence Balter (20th century)

    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)

    Men sometimes speak as if the study of the classics would at length make way for more modern and practical studies; but the adventurous student will always study classics, in whatever language they may be written and however ancient they may be. For what are the classics but the noblest recorded thoughts of man?... We might as well omit to study Nature because she is old.
    Henry David Thoreau (1817–1862)

    There is a potential 4-6 percentage point net gain for the President [George Bush] by replacing Dan Quayle on the ticket with someone of neutral stature.
    Mary Matalin, U.S. Republican political advisor, author, and James Carville b. 1946, U.S. Democratic political advisor, author. All’s Fair: Love, War, and Running for President, p. 205, Random House (1994)