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 M 200000
#define SNT 2000000000
typedef long long int llong;

int L[M / 2 + 2], R[M / 2 + 2];

llong merge(int*, int, int, int, int);
llong mergeSort(int*, int, int, int);

llong merge(int A[], int n, int left, int mid, int right){
  int i, j, k;
  int n1, n2;
  llong count = 0;
  n1 = mid - left;
  n2 = right - mid;
  for( i = 0 ; i < n1 ; i++){
    L[i] = A[left + i];
  }
  for( i = 0 ; i < n2 ; i++){
    R[i] = A[mid + i];
  }
  L[n1] = R[n2] = SNT;
  i = 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;
}

llong mergeSort(int A[], int n, int left, int right){
  int mid;
  llong v1, v2, v3;
  if( left + 1 < right){
    mid = (left + right) / 2;
    v1 = mergeSort(A, n, left, mid);
    v2 = mergeSort(A, n, mid, right);
    v3 = merge(A, n, left, mid, right);
    return v1 + v2 + v3;
  }
  else return 0;
}

int main(){
  int A[M], n, i;
  llong ans;

  scanf("%d", &n);
  for( i = 0 ; i < n ; i++){
    scanf("%d", &A[i]);
  }

  ans = mergeSort(A, n, 0, n);
  printf("%lld\n", ans);

  return 0;
}