Binary Inversions
Description
You are given a binary array$^{\dagger}$ of length $n$. You are allowed to perform one operation on it at most once. In an operation, you can choose any element and flip it: turn a $0$ into a $1$ or vice-versa.
What is the maximum number of inversions$^{\ddagger}$ the array can have after performing at most one operation?
$^\dagger$ A binary array is an array that contains only zeroes and ones.
$^\ddagger$ The number of inversions in an array is the number of pairs of indices $i,j$ such that $i<j$ and $a_i > a_j$.
The input consists of multiple test cases. The first line contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer $n$ ($1 \leq n \leq 2\cdot10^5$) — the length of the array.
The following line contains $n$ space-separated positive integers $a_1$, $a_2$,..., $a_n$ ($0 \leq a_i \leq 1$) — the elements of the array.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot10^5$.
For each test case, output a single integer — the maximum number of inversions the array can have after performing at most one operation.
Input
The input consists of multiple test cases. The first line contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains an integer $n$ ($1 \leq n \leq 2\cdot10^5$) — the length of the array.
The following line contains $n$ space-separated positive integers $a_1$, $a_2$,..., $a_n$ ($0 \leq a_i \leq 1$) — the elements of the array.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot10^5$.
Output
For each test case, output a single integer — the maximum number of inversions the array can have after performing at most one operation.
5
4
1 0 1 0
6
0 1 0 0 1 0
2
0 0
8
1 0 1 1 0 0 0 1
3
1 1 1
3
7
1
13
2
Note
For the first test case, the inversions are initially formed by the pairs of indices ($1, 2$), ($1, 4$), ($3, 4$), being a total of $3$, which already is the maximum possible.
For the second test case, the inversions are initially formed by the pairs of indices ($2, 3$), ($2, 4$), ($2, 6$), ($5, 6$), being a total of four. But, by flipping the first element, the array becomes ${1, 1, 0, 0, 1, 0}$, which has the inversions formed by the pairs of indices ($1, 3$), ($1, 4$), ($1, 6$), ($2, 3$), ($2, 4$), ($2, 6$), ($5, 6$) which total to $7$ inversions which is the maximum possible.