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

“In many ways, life becomes simpler [for young adults]. . . . We are expected to solve only a finite number of problems within a limited range of possible solutions. . . . It’s a mental vacation compared with figuring out who we are, what we believe, what we’re going to do with our talents, how we’re going to solve the social problems of the globe . . .and what the perfect way to raise our children will be.”
—Roger Gould (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)

“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)

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