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Proof of the Rencontres numbers

  1. Nov 18, 2011 #1
    Hi everyone,

    I'm reading this article

    http://en.wikipedia.org/wiki/Rencontres_numbers

    and although I understand the whole concept of these numbers, I can't understand how the author derives this equation

    [PLAIN]http://img809.imageshack.us/img809/529/unledid.png [Broken]

    this recursion essentially counts the amount of permutation where no number is at a fixed point in the array of n+2 size

    I can count these things for small n's for example let's say n is 3

    we have these permuations

    1 2 3 -> 3
    1 3 2 -> 1
    3 1 2 -> 0
    2 3 1 -> 0
    2 1 3 -> 1

    thus the amount we're looking for is 2

    but what about any n? this is what the author is calculating, I've spent many hours already and I can't figure out where does this recursion come from and how exactly does he go from the recursion to the final equation which is the following:

    [URL]http://upload.wikimedia.org/wikipedia/en/math/d/6/a/d6a38c5b9f46d83ef8310b4288aebe3d.png[/URL]

    thanks in advance :)
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
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