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:

    “For me chemistry represented an indefinite cloud of future potentialities which enveloped my life to come in black volutes torn by fiery flashes, like those which had hidden Mount Sinai. Like Moses, from that cloud I expected my law, the principle of order in me, around me, and in the world.... I would watch the buds swell in spring, the mica glint in the granite, my own hands, and I would say to myself: “I will understand this, too, I will understand everything.””
    —Primo Levi (1919–1987)

    “A child’s self-image is more like a scrapbook than a single snapshot. As the child matures, the number and variety of images in that scrapbook may be far more important than any individual picture pasted inside it.”
    —Lawrence Kutner (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)

    “One writes of scars healed, a loose parallel to the pathology of the skin, but there is no such thing in the life of an individual. There are open wounds, shrunk sometimes to the size of a pin-prick but wounds still. The marks of suffering are more comparable to the loss of a finger, or the sight of an eye. We may not miss them, either, for one minute in a year, but if we should there is nothing to be done about it.”
    —F. Scott Fitzgerald (1896–1940)