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In http://www.springerlink.com/content/y19u81675243r237/fulltext.pdf, the author states the following without proof (equation 3.1):

Consider a random permutation $\pi$ of $\mathbb{Z}_n$. What is the probability that $\pi(i+1)-\pi(i) \pmod{n} <n/2$ for all $i$?

The claim is that this is $(2+o(1))^{-n}$, which makes sense and seems like it should be a standard argument. Does anyone have a formal proof?

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The maximum $\pi(i+1)- \pi(i)$ can be is $n-1$ in $\mathbb{Z}_n$. –  PEV Jan 31 '11 at 23:09
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I think you mean (2 + o(1))^{-n}? –  Qiaochu Yuan Jan 31 '11 at 23:36
    
Good call, Qiaochu. Fixed. –  Jeremy Hurwitz Feb 1 '11 at 6:21
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1 Answer

up vote 2 down vote accepted

This question was answered at MathOverflow: http://mathoverflow.net/questions/54051/random-permutations-of-z-n

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