A move swaps $x=p_i$ and $y=p_j$ with $i<j$ and $x>y$, earning $j-i$ points.

Consider the weighted sum

$S(p)=\sum_{i=1}^{n}i p_i.$

Only two terms change in a swap. Its increase is

$iy+jx-ix-jy=(j-i)(x-y).$

Since $x-y\ge1$, this increase is at least the points earned.

So every earned point must be paid for by an increase in $S$. If we bound how far $S$ can increase, we also bound the total score.

Equality holds for a move exactly when its two values differ by $1$. We will use that later.