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.