Intuition: if f(a) >= f(b), then k(a) <= k(b) so that total # is minimized

Proof: Suppose there exists f(a) >= f(b), and k(a) > k(b), swapping the key of a and b will reduce the total #. Thus, we can repeat the process and keep reducing the total #,i.e., an arrangement satifies the condition is better than all arrangments not satisfying the condition, this means the condition is optimal

Note that sorting is increasing order by default. In this case we need decreasing order, or we can start checking from tail to head instead of head to tail