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Let $G=(V,E)$ be a graph. The gradient of a function $f:V\longrightarrow R$ is defined on the edges of the graph, given by the discrete derivative $ \nabla f(e)=f(y)-f(x), \quad e=(x,y) \in E $

Take $V=S_{2n}$ the group of permutations $\pi$ of the set $\{1, \ldots 2n\}$ and $e=(\pi, \pi \tau) \in E$ for some transposition $\tau$. Consider $f=\left|\sum\limits_{i=1}^na_{\pi(i)}-\sum\limits_{i=n+1}^{2n}a_{\pi(i)}\right|$.

Find $\|\nabla f\|_{\infty}$.

I wasn't sure, but it turns out that this is kind of rephrased question as here question involving Markov chain Thank you.

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As already explained here, that is, on a page you know and should have mentioned, $ \|\nabla f\|_\infty\leqslant2\cdot\max\{|a_i-a_j|\,;\,1\leqslant i\leqslant n,\,n+1\leqslant j\leqslant 2n\}. $

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    I've already got an answer for my question. I just was confused with the typo in the answer above. Thank you very much for your help. I really appreciate it:-)2012-08-03