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;
long long n;
struct BIT {
  long long c[100009];
  void add(long long x, long long y) {
    for (; x <= n; x += (x & -x)) c[x] += y;
  }
  long long sum(long long x) {
    long long s = 0;
    for (; x; x -= (x & -x)) s += c[x];
    return s;
  }
} C, S;
double ans;
long long a[100009];
signed main() {
  cin >> n;
  for (long long i = 1; i <= n; i++) cin >> a[i];
  for (long long i = 1; i <= n; i++) ans += 0.5 * i * (n - i) * (n - i + 1);
  for (long long i = n; i; i--) {
    ans += C.sum(a[i] - 1) * (n * n + n - 2 * i - 2 * n * i);
    ans += 2 * S.sum(a[i] - 1) * i;
    C.add(a[i], 1);
    S.add(a[i], i);
  }
  cout << setprecision(20) << fixed << 1.0 * ans / n / (n + 1);
}