CF1637H.Minimize Inversions Number

You are given a permutation $p$ of length $n$ .

You can choose any subsequence, remove it from the permutation, and insert it at the beginning of the permutation keeping the same order.

For every $k$ from $0$ to $n$ , find the minimal possible number of inversions in the permutation after you choose a subsequence of length exactly $k$ .

The first line contains a single integer $t$ ( $1 \le t \le 50\,000$ ) — the number of test cases.

The first line of each test case contains one integer $n$ ( $1 \le n \le 5 \cdot 10^5$ ) — the length of the permutation.

The second line of each test case contains the permutation $p_1, p_2, \ldots, p_n$ ( $1 \le p_i \le n$ ).

It is guaranteed that the total sum of $n$ doesn't exceed $5 \cdot 10^5$ .

For each test case output $n + 1$ integers. The $i$ -th of them must be the answer for the subsequence length of $i - 1$ .

In the second test case:

• For the length $0$ : $[4, 2, 1, 3] \rightarrow [4, 2, 1, 3]$ : $4$ inversions.
• For the length $1$ : $[4, 2, \mathbf{1}, 3] \rightarrow [1, 4, 2, 3]$ : $2$ inversions.
• For the length $2$ : $[4, \mathbf{2}, \mathbf{1}, 3] \rightarrow [2, 1, 4, 3]$ , or $[4, 2, \mathbf{1}, \textbf{3}] \rightarrow [1, 3, 4, 2]$ : $2$ inversions.
• For the length $3$ : $[4, \mathbf{2}, \mathbf{1}, \mathbf{3}] \rightarrow [2, 1, 3, 4]$ : $1$ inversion.
• For the length $4$ : $[\mathbf{4}, \mathbf{2}, \mathbf{1}, \mathbf{3}] \rightarrow [4, 2, 1, 3]$ : $4$ inversions.