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Thread: Random Permutation

  1. #1
    Senior Member Sampras's Avatar
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    Random Permutation

    Suppose $\displaystyle \pi $ is a random permutation of $\displaystyle \mathbb{Z}_n $. How would you determine the probability that $\displaystyle \pi(i+1)-\pi(i) \pmod n < n/2 $? We know that $\displaystyle \pi(i+1)-\pi(i) $ is maximum at $\displaystyle n-1 $ in $\displaystyle \mathbb{Z}_n $.
    Last edited by Sampras; Jan 31st 2011 at 03:11 PM.
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  2. #2
    Senior Member Sampras's Avatar
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    Perhaps we could use Inclusion-Exclusion? In other words, find the probability that $\displaystyle \pi(i+1)-\pi(i) > n/2 $ (i.e. the $\displaystyle i $th and $\displaystyle i+1 $-st places differ by more than $\displaystyle n/2 $?
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