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Let vector $c\in 2N $ is such that first $m$ of its coordinates are $1$ and the rest are $0$ ($c=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2N\}$. Define $$g=\left|\sum_{i=1}^N c_{\pi(i)}-\sum_{i=N+1}^{2N}c_{\pi(i)}\right|.$$

I would like to calculate or approximate expectation $E|g|^q,$ for any $q\ge 2$.

Any help would be appreciated.

Thank you.

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1. What do you mean by $k$-th permutation? 2. What does $E|g|^p$ mean? 3. Just to make sure that I understand it: If $\pi$ would be the identity and $m\le N$, then $g=m$, right? – draks ... Apr 22 '12 at 21:17
Did you try for $q=2$? – Did Apr 22 '12 at 22:04
@Didier: Yes, for $q=2$ is simple. I would like the upper bound $(m)^{q/2}$. If I split the sum (in the representation of the expectation) into small intervals of summation, say $k\in [m/2-C\sqrt m; m/2+C\sqrt m]$, (with $C$-constant) then it is hard to estimate the pats $k\in [0, m/2-C\sqrt m]$ and $k\in [m/2+C\sqrt m, m]$. – David Apr 22 '12 at 22:14
You might want to show the computations when $q=2$. – Did Apr 22 '12 at 22:26
What happened with my last suggestion? – Did May 23 '12 at 21:00

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