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.BufferedReader;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.StringTokenizer;

public class E {

	static class FenwickTree {
		long[] t;
		public FenwickTree(int n) {
			t=new long[n];
		}
		long get(int r) {
			long res = 0;
			for (int i = r; i >= 0; i &= i + 1, --i) res += t[i];
			return res;
		}
		long get(int l, int r) {
			return get(r) - get(l - 1);
		}
		void update(int pos, long val) {
			for (int i = pos; i < t.length; i |= i + 1) t[i] += val;
		}
	}
	
	public static void main(String[] args) throws Exception {
		FastScanner scan = new FastScanner();
		PrintWriter out = new PrintWriter(System.out);
		int n = scan.nextInt();
		int[] a = new int[n];
		for(int i = 0; i < n; i++) a[i] = scan.nextInt()-1;
		FenwickTree b = new FenwickTree(n);
		FenwickTree c = new FenwickTree(n);
		double p = 0;
		for(int i = 0; i < n; i++){
			p += b.get(0, n-1)*1.0*(n-i)/n/(n+1);
			p += c.get(a[i]+1, n-1);
			p -= b.get(a[i]+1, n-1)*1.0*(n-i)/n/(n + 1)*2;
			c.update(a[i], 1);
			b.update(a[i], i+1);
		}
		out.println(p);
		out.close();
	}

	static class FastScanner {
		BufferedReader br;
		StringTokenizer st;

		public FastScanner() {
			try	{
				br = new BufferedReader(new InputStreamReader(System.in));
				st = new StringTokenizer(br.readLine());
			} catch (Exception e){e.printStackTrace();}
		}

		public String next() {
			if (st.hasMoreTokens())	return st.nextToken();
			try {st = new StringTokenizer(br.readLine());}
			catch (Exception e) {e.printStackTrace();}
			return st.nextToken();
		}

		public int nextInt() {return Integer.parseInt(next());}

		public long nextLong() {return Long.parseLong(next());}

		public double nextDouble() {return Double.parseDouble(next());}

		public String nextLine() {
			String line = "";
			try {line = br.readLine();} 
			catch (Exception e) {e.printStackTrace();}
			return line;
		}

		public int[] nextIntArray(int n) {
			int[] a = new int[n];
			for(int i = 0; i < n; i++) a[i] = nextInt();
			return a;
		}

		public long[] nextLongArray(int n){
			long[] a = new long[n];
			for(int i = 0; i < n; i++) a[i] = nextLong();
			return a;
		}

		public double[] nextDoubleArray(int n){
			double[] a = new double[n];
			for(int i = 0; i < n; i++) a[i] = nextDouble();
			return a;
		}

		public char[][] nextGrid(int n, int m){
			char[][] grid = new char[n][m];
			for(int i = 0; i < n; i++) grid[i] = next().toCharArray();
			return grid;
		}
	}

}