CF1730F.Almost Sorted

You are given a permutation $p$ of length $n$ and a positive integer $k$ . Consider a permutation $q$ of length $n$ such that for any integers $i$ and $j$ , where $1 \le i < j \le n$ , we have $$$$p_{q_i} \le p_{q_j} + k. $$

Find the minimum possible number of inversions in a permutation $q$ .

A permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, \[2,3,1,5,4\] is a permutation, but \[1,2,2\] is not a permutation ( $2$ appears twice in the array) and \[1,3,4\] is also not a permutation ( $n=3$ but there is $4$ in the array).

An inversion in a permutation $a$ is a pair of indices $i$ and $j$ ( $1 \\le i, j \\le n$ ) such that i < j , but a\_i > a\_j$$.

The first line contains two integers $n$ and $k$ ( $1 \le n \le 5000$ , $1 \le k \le 8$ ).

The second line contains $n$ distinct integers $p_1, p_2, \ldots, p_n$ ( $1 \le p_i \le n$ ).

Print the minimum possible number of inversions in the permutation $q$ .

In the first example, the only permutation is $q = [1]$ ( $0$ inversions). Then $p_{q_1} = 1$ .

In the second example, the only permutation with $1$ inversion is $q = [1, 3, 2]$ . Then $p_{q_1} = 2$ , $p_{q_2} = 1$ , $p_{q_3} = 3$ .

In the third example, one of the possible permutations with $6$ inversions is $q = [3, 4, 5, 1, 2]$ . Then $p_{q_1} = 3$ , $p_{q_2} = 2$ , $p_{q_3} = 1$ , $p_{q_4} = 5$ , $p_{q_5} = 4$ .