Monk and Inversions

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Data Structures

Details

Problem

Monk's best friend Micro, who happen to be an awesome programmer, got him an integer matrix M of size $N \times N$ for his birthday. Monk is taking coding classes from Micro. They have just completed array inversions and Monk was successful in writing a program to count the number of inversions in an array. Now, Micro has asked Monk to find out the number of inversion in the matrix M. Number of inversions, in a matrix is defined as the number of unordered pairs of cells $\{(i, j), (p, q)\}$ such that $M[i][j] \gt M[p][q]\space \&\space i \le p\space \& \space j \le q$ .
Monk is facing a little trouble with this task and since you did not got him any birthday gift, you need to help him with this task.

Video approach to solve this question: here

Input:
First line consists of a single integer T denoting the number of test cases.
First line of each test case consists of one integer denoting N. Following N lines consists of N space separated integers denoting the matrix M.

Output:
Print the answer to each test case in a new line.

Constraints:
$1 \le T \le 100$
$1 \le N \le 20$
$1 \le M[i][j] \le 1000$

Sample Input

Sample Output

0
2

Time Limit: 1

Memory Limit: 256

Source Limit:

Explanation

In first test case there is no pair of cells $(x_1, y_1)$ , $(x_2, y_2)$ , $x_1 \le x_2 \space \& \space y_1 \le y_2$ having $M[x_1][y_1] > M[x_2][y_2]$ , so the answer is 0.
In second test case $M[1][1] > M[1][2]$ and $M[1][1] \gt M[2][1]$ , so the answer is 2.

Contributers:

Vaibhav Jaimini