Let $a_i$ be $n$ distinct real numbers. What is the expectation:
$$\mathbb E_\sigma \left[ \sum_{i=1}^{n} \frac {1} {a_{\sigma(i)} - \sum_{j=1}^{i-1}a_{\sigma(j)}} \right] $$
where the expectation is taken over all $n!$ permutations $\sigma:[1,2,..,n]\rightarrow[1,2,..,n]$
I appreciate your help.