Random Permutation Statistics - Probability That A Random Subset of Lies On The Same Cycle

Probability That A Random Subset of Lies On The Same Cycle

Select a random subset Q of containing m elements and a random permutation, and ask about the probability that all elements of Q lie on the same cycle. This is another average parameter. The function b(k) is equal to, because a cycle of length k contributes subsets of size m, where for k < m. This yields

\frac{\partial}{\partial u} g(z, u) \Bigg|_{u=1} = \frac{1}{1-z} \sum_{k\ge m} {k \choose m} \frac{z^k}{k} = \frac{1}{1-z} \frac{1}{m} \frac{z^m}{(1-z)^m} = \frac{1}{m} \frac{z^m}{(1-z)^{m+1}}.

Averaging out we obtain that the probability of the elements of Q being on the same cycle is

{n \choose m}^{-1} \frac{1}{m} \frac{z^m}{(1-z)^{m+1}} = {n \choose m}^{-1} \frac{1}{m} \frac{1}{(1-z)^{m+1}}

or

\frac{1}{m} {n \choose m}^{-1} {(n-m) \; + \; m \choose m} = \frac{1}{m}.

In particular, the probability that two elements p < q are on the same cycle is 1/2.

Read more about this topic:  Random Permutation Statistics

Famous quotes containing the words probability, random, lies and/or cycle:

    “The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
    —Andrew Michael Ramsay (1686–1743)

    “Novels as dull as dishwater, with the grease of random sentiments floating on top.”
    —Italo Calvino (1923–1985)

    “So live that when thy summons comes to join
    The innumerable caravan that moves
    To that mysterious realm, where each shall take
    His chamber in the silent halls of death,
    Thou go not, like the quarry-slave at night,
    Scourged to his dungeon, but, sustained and soothed
    By an unfaltering trust, approach thy grave
    Like one who wraps the drapery of his couch
    About him and lies down to pleasant dreams.”
    —William Cullen Bryant (1794–1878)

    “Only mediocrities progress. An artist revolves in a cycle of masterpieces, the first of which is no less perfect than the last.”
    —Oscar Wilde (1854–1900)