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;
const int maxn = 1e5 + 10;
long long tree[maxn][2];
int a[maxn];
long long n;
int lowbit(int x) { return x & (-x); }
void add(int i, int val, int flag) {
  while (i <= n) {
    tree[i][flag] += val;
    i += lowbit(i);
  }
}
long long query(int x, int flag) {
  long long ans = 0;
  while (x) {
    ans += tree[x][flag];
    x -= lowbit(x);
  }
  return ans;
}
int main() {
  memset(tree, 0, sizeof(tree));
  scanf("%lld", &n);
  for (long long i = 1; i <= n; i++) {
    scanf("%d", &a[i]);
  }
  double res = 0;
  for (long long i = 1; i <= n; i++) {
    res += i * (n - i) * (n - i + 1) / 2;
  }
  for (long long i = n; i >= 1; i--) {
    res += (n * n + n - 2 * n * i - 2 * i) * query(a[i] - 1, 0);
    res += 2 * i * query(a[i] - 1, 1);
    add(a[i], 1, 0);
    add(a[i], i, 1);
  }
  res /= (n + 1) * n;
  printf("%.12lf\n", res);
}