Problem: You are given a permutation of integers from 1 to n. Exactly once you apply the following operation to this permutation: pick a random segment and shuffle its elements. Formally:

  1. Pick a random segment (continuous subsequence) from l to r. All <image> segments are equiprobable. 
  2. Let k = r - l + 1, i.e. the length of the chosen segment. Pick a random permutation of integers from 1 to k, p1, p2, ..., pk. All k! permutation are equiprobable. 
  3. This permutation is applied to elements of the chosen segment, i.e. permutation a1, a2, ..., al - 1, al, al + 1, ..., ar - 1, ar, ar + 1, ..., an is transformed to a1, a2, ..., al - 1, al - 1 + p1, al - 1 + p2, ..., al - 1 + pk - 1, al - 1 + pk, ar + 1, ..., an. 



Inversion if a pair of elements (not necessary neighbouring) with the wrong relative order. In other words, the number of inversion is equal to the number of pairs (i, j) such that i < j and ai > aj. Find the expected number of inversions after we apply exactly one operation mentioned above.

Input

The first line contains a single integer n (1 ≤ n ≤ 100 000) — the length of the permutation.

The second line contains n distinct integers from 1 to n — elements of the permutation.

Output

Print one real value — the expected number of inversions. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 9. 

Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if <image>.

Example

Input

3
2 3 1


Output

1.916666666666666666666666666667

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
int n, a[100007];
long long bit[100007];
long long f[100007];
double res;
void update_bit(int x, int v) {
  while (x > 0) {
    bit[x] += v;
    x -= (x & (-x));
  }
}
long long get_bit(int x) {
  long long res = 0;
  while (x <= n) {
    res += bit[x];
    x += (x & (-x));
  }
  return res;
}
int main() {
  cout.precision(9);
  cout << fixed;
  cin >> n;
  for (int i = 1; i <= n; ++i) cin >> a[i];
  f[1] = 0;
  for (int i = 2; i <= n; ++i) f[i] = f[i - 1] + (i - 1);
  long long r = 0;
  for (int i = 1; i <= n; ++i) r = r + f[i] * (n + 1 - i);
  for (int i = 1; i <= n; ++i) {
    r -= get_bit(a[i]) * 2 * (n + 1 - i);
    update_bit(a[i], i);
  }
  res = r;
  res = res / n / (n + 1);
  memset(bit, 0, sizeof(bit));
  for (int i = 1; i <= n; ++i) {
    res += get_bit(a[i]);
    update_bit(a[i], 1);
  }
  cout << res;
  return 0;
}