Let $n$ be a positive integer and let $(a_1,...,a_n)$ be a permutation of {$1,2,...,n$}.
Define
$A_k$ = {$a_i | a_i < a_k, i > k$} and $B_k$ = {$a_i | a_i > a_k, i<k$} for $1$$\leq$$k$$\leq$$n$.
Prove that $$\sum_{k=1}^{n} |A_k|=\sum_{k=1}^{n} |B_k|$$
What idea would be useful? I though that using sign in permutation would be useful, but I can't develop proof based on that.