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 = 100000 + 10;
int N, A[maxn];
long long K = 0ll, t = 0ll, T = 0ll;
double E = 0.0;
struct Fenwick_Tree {
  long long val[maxn];
  int lowbit(int x) { return x & -x; }
  void Update(int p, long long v) {
    for (int i = p; i <= N; i += lowbit(i)) val[i] += v;
  }
  long long Sum(int p) {
    long long tot = 0ll;
    for (int i = p; i; i -= lowbit(i)) tot += val[i];
    return tot;
  }
} T1, T2;
void Init() {
  scanf("%d", &N);
  for (int i = 1; i <= N; ++i) scanf("%d", &A[i]), A[i] = N - A[i] + 1;
}
void Solve() {
  if (N == 1) {
    printf("0\n");
    return;
  }
  for (int i = 1; i <= N; ++i) {
    K += T1.Sum(A[i]), T1.Update(A[i], 1);
    t += T2.Sum(A[i]), T2.Update(A[i], i);
    T += t, E += 1.0 * i * (i - 1) / 4.0 * (N - i + 1);
  }
  double ans = K + (double)(E - T) * 2.0 / N / (N + 1);
  printf("%.15lf\n", ans);
}
int main() {
  Init(), Solve();
  return 0;
}