Random Permutation Statistics - Expected Number of Cycles of A Given Size m

Expected Number of Cycles of A Given Size m

In this problem we use a bivariate generating function g(z, u) as described in the introduction. The value of b for a cycle not of size m is zero, and one for a cycle of size m. 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} \frac{z^m}{m}

or

\frac{1}{m} z^m \; + \; \frac{1}{m} z^{m+1} \; + \; \frac{1}{m} z^{m+2} \; + \; \cdots

This means that the expected number of cycles of size m in a permutation of length n less than m is zero (obviously). A random permutation of length at least m contains on average 1/m cycles of length m. In particular, a random permutation contains about one fixed point.

The OGF of the expected number of cycles of length less than or equal to m is therefore

\frac{1}{1-z} \sum_{k=1}^m \frac{z^k}{k} \mbox{ and } \frac{1}{1-z} \sum_{k=1}^m \frac{z^k}{k} = H_m \mbox{ for } n \ge m

where Hm is the mth harmonic number. Hence the expected number of cycles of length at most m in a random permutation is about ln m.

Read more about this topic:  Random Permutation Statistics

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

    “I believe that all women of working ages and physical capacity, regardless of income, should be expected to earn their livings either in or out of the home. Until this attitude prevails I believe the position of women will be uncertain and undignified, in spite of poetic rhapsodies to the contrary.”
    —Mary Barnett Gilson (1877–?)

    “The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language.”
    —Anton Pavlovich Chekhov (1860–1904)

    “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)

    “Learn to shrink yourself to the size of the company you are in. Take their tone, whatever it may be, and excell in it if you can; but never pretend to give the tone. A free conversation will no more bear a dictator than a free government will.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)