Problem:
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The Number of Inversions

For a given sequence $A = \{a_0, a_1, ... a_{n-1}\}$, the number of pairs $(i, j)$ where $a_i > a_j$ and $i < j$, is called the number of inversions. The number of inversions is equal to the number of swaps of Bubble Sort defined in the following program:

bubbleSort(A)
cnt = 0 // the number of inversions
for i = 0 to A.length-1
for j = A.length-1 downto i+1
if A[j] < A[j-1]
swap(A[j], A[j-1])
cnt++

return cnt

For the given sequence $A$, print the number of inversions of $A$. Note that you should not use the above program, which brings Time Limit Exceeded.

Input

In the first line, an integer $n$, the number of elements in $A$, is given. In the second line, the elements $a_i$ ($i = 0, 1, .. n-1$) are given separated by space characters.

output

Print the number of inversions in a line.

Constraints

$ 1 \leq n \leq 200,000$

$ 0 \leq a_i \leq 10^9$

$a_i$ are all different

Sample Input 1

5
3 5 2 1 4

Sample Output 1

6

Sample Input 2

3
3 1 2

Sample Output 2

2

Solution in C (judge verdict: Accepted)

#include <stdio.h>
#define MAX 200000
#define SENTINEL 2000000000
long long merge(int,int,int);
long long mergeSort(int,int);
int A[MAX];
int L[MAX / 2 + 2],R[MAX / 2 + 2];

int main(){
  int n,i;
  long long result;
  scanf("%d",&n);
  for(i = 0; i < n; i++){
    scanf("%d",&A[i]);
  }
  result = mergeSort(0,n);
  printf("%lu\n",result);
  return 0;
}

long long merge(int left,int mid,int right){
  long long count = 0;
  int i,j,k;
  int n1,n2;
  n1 = mid - left;
  n2 = right - mid;
  for(i = 0; i < n1; i++) L[i] = A[left + i];
  for(j = 0; j < n2; j++) R[j] = A[mid + j];
  L[n1] = SENTINEL;
  R[n2] = SENTINEL;
  i = 0;
  j = 0;
  for(k = left; k < right; k++){
    if(L[i] <= R[j]){
      A[k] = L[i++];
    }
    else{
      A[k] = R[j++];
      count += n1 - i;
    }
  }
  return count;
}

long long mergeSort(int left, int right){
  int mid;
  long long a1,a2,a3;
  if(left + 1 < right){
    mid = (left + right)/2;
    a1 = mergeSort(left,mid);
    a2 = mergeSort(mid,right);
    a3 = merge(left,mid,right);
    return a1 + a2 + a3;
  }
  else return 0;
}