Sorting a Sequence 5

Let $S$ be the initial sortedness and $D$ the number of strict inversions. Swapping an adjacent inversion increases sortedness by one. If $K\leq D$, perform $K$ insertion-sort swaps to attain $S+K$. Otherwise sort first. If duplicates exist, repeatedly swap an adjacent equal pair for the remaining operations. If all elements differ, spend pairs of operations swapping the same position twice; an odd remainder loses one unit of sortedness. Permutation parity makes this loss unavoidable.

Time is $O(N^2+K)$ and sortedness is at most $499500$. The small subtask can enumerate operation sequences with $O(1)$ score updates per swap. The checker requires exactly $K$ valid indices and awards only for a correct optimum with a nonoptimal, properly formatted witness.