Almost Sorted
Description
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 $q$.
Input
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 $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$).
Output
Print the minimum possible number of inversions in the permutation $q$.
1 1
1
3 1
2 3 1
5 2
5 4 3 2 1
10 3
5 8 6 10 2 7 4 1 9 3
0
1
6
18
Note
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 $p_{q_1} = 3$, $p_{q_2} = 2$, $p_{q_3} = 1$, $p_{q_4} = 5$, $p_{q_5} = 4$.