CF1760E.Binary Inversions

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.

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.