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++11 (judge verdict: Wrong Answer)

#include<iostream>
#include<vector>
using namespace std;

int merge_sort(vector<int>& A, int l, int r) {
    int cnt = 0;
    if (l >= r) return 0;
    int m = (l + r) / 2;
    cnt += merge_sort(A, l    , m);
    cnt += merge_sort(A, m + 1, r);
    vector<int> B1(m - l + 2);
    for (int i = 0; i < m - l + 1; i++) B1[i] = A[i+l];
    B1[m-l+1] = 1e9 + 1;
    vector<int> B2(r - m + 1);
    for (int i = 0; i < r - m; i++) B2[i] = A[i+m+1];
    B2[r-m] = 1e9 + 1;
    int j1 = 0;
    int j2 = 0;
    for (int i = l; i <= r; i++) {
        if (B1[j1] <= B2[j2]) {
            A[i] = B1[j1++];
        } else {
            A[i] = B2[j2++];
            cnt += m - l + 1 - j1;
        }
    }
    return cnt;
}

int main(){
    int n;
    cin >> n;
    vector<int> A(n);
    for (auto& a : A) cin >> a;
    cout << merge_sort(A, 0, n - 1) << endl;
    return 0;
}