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 JAVA):

Solution:
import java.io.*;
import java.util.*;

public class CF749E {
	// Fenwick tree
	static void update(long[] aa, int i, long a) {
		while (i < aa.length) {
			aa[i] += a;
			i |= i + 1;
		}
	}
	static long query(long[] aa, int i) {
		long sum = 0;
		while (i >= 0) {
			sum += aa[i];
			i &= i + 1;
			i--;
		}
		return sum;
	}
	static class A {
		int a, i;
		A(int a, int i) {
			this.a = a;
			this.i = i;
		}
	}
	public static void main(String[] args) throws IOException {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
		int n = Integer.parseInt(br.readLine());
		StringTokenizer st = new StringTokenizer(br.readLine());
		A[] aa = new A[n];
		for (int i = 0; i < n; i++) {
			int a = Integer.parseInt(st.nextToken());
			aa[i] = new A(a, i);
		}
		Arrays.sort(aa, (p, q) -> q.a - p.a);
		long m = (long) n * (n + 1) / 2;
		double inv = 0;
		long[] tt = new long[n];
		for (int i = 0; i < n; i++) {
			int i_ = aa[i].i;
			inv += query(tt, i_) * m;
			update(tt, i_, 1);
		}
		Arrays.fill(tt, 0);
		for (int i = 0; i < n; i++) {
			int i_ = aa[i].i;
			inv -= query(tt, i_) * (n - i_);
			update(tt, i_, i_ + 1);
		}
		// expected number of inversions in an i-permutation is (i choose 2) / 2
		// n - i + 1 shifted i-intervals in an n-interval
		for (int i = 1; i <= n; i++)
			inv += (long) (n - i + 1) * i * (i - 1) / 4.0;
		System.out.println(inv / m);
	}
}