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 $ .