Sorting with Set

For a permutation $P$, define $f(P, K)$ in the following way: - Start with an empty set of pairs $S$. - You can do the following action repeatedly as many times as you want: - Choose $2$ integers $1 \le u < v \le N$ - Swap $P_u$ and $P_v$ - Add $(u, v)$ to our set $S$ - When $P$ is sorted, we define the $score$ to be $\sum(u + v + K)$ for distinct $(u, v)$ in the set $S$ - $f(P)$