A15648 | Lucky Permutation

You are given a permutation $^\dagger$ $p$ of length $n$ .

In one operation, you can choose two indices $1 \le i < j \le n$ and swap $p_i$ with $p_j$ .

Find the minimum number of operations needed to have exactly one inversion $^\ddagger$ in the permutation.

$^\dagger$ 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).

$^\ddagger$ The number of inversions of a permutation $p$ is the number of pairs of indices $(i, j)$ such that $1 \le i < j \le n$ and $p_i > p_j$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).

The second line of each test case contains $n$ integers $p_1,p_2,\ldots, p_n$ ( $1 \le p_i \le n$ ). It is guaranteed that $p$ is a permutation.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case output a single integer — the minimum number of operations needed to have exactly one inversion in the permutation. It can be proven that an answer always exists.