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

/**
 * @author Don Li
 */
public class InversionsAfterShuffle {
    
    void solve() {
        int n = in.nextInt();
        int[] a = new int[n];
        for (int i = 0; i < n; i++) a[i] = in.nextInt();
        
        int[] pos = new int[n + 1];
        for (int i = 0; i < n; i++) pos[a[i]] = i;
        
        long[] sum = new long[n + 1];
        sum[n] = 0;
        for (int i = n - 1; i >= 0; i--) sum[i] = sum[i + 1] + (n - i);
    
        double ans = 0;
        long tot = (long) n * (n + 1) / 2;
        Fenwick fenwickSum = new Fenwick(n), fenwickCnt = new Fenwick(n);
        for (int i = 1; i <= n; i++) {
            int j = pos[i];
            
            long less = fenwickCnt.sum(j + 1, n - 1), more = n - j - 1 - less;
            long lessSum = fenwickSum.sum(j + 1, n - 1), moreSum = sum[j + 1] - lessSum;
            ans += less - 0.5 * lessSum * (j + 1) / tot;
            ans += 0.5 * moreSum * (j + 1) / tot;
            
            fenwickSum.add(j, n - j);
            fenwickCnt.add(j, 1);
        }
        out.println(ans);
    }
    
    static class Fenwick {
        long[] d;
        
        public Fenwick(int n) {
            d = new long[n];
        }
        
        long sum(int i) {
            long ret = 0;
            for (int x = i; x >= 0; x = (x & (x + 1)) - 1) {
                ret += d[x];
            }
            return ret;
        }
        
        void add(int i, long delta) {
            for (int x = i; x < d.length; x |= x + 1) {
                d[x] += delta;
            }
        }
        
        long sum(int l, int r) {
            return sum(r) - sum(l - 1);
        }
    }
    
    public static void main(String[] args) {
        in = new FastScanner(new BufferedReader(new InputStreamReader(System.in)));
        out = new PrintWriter(System.out);
        new InversionsAfterShuffle().solve();
        out.close();
    }
    
    static FastScanner in;
    static PrintWriter out;
    
    static class FastScanner {
        BufferedReader in;
        StringTokenizer st;
        
        public FastScanner(BufferedReader in) {
            this.in = in;
        }
        
        public String nextToken() {
            while (st == null || !st.hasMoreTokens()) {
                try {
                    st = new StringTokenizer(in.readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }
        
        public int nextInt() {
            return Integer.parseInt(nextToken());
        }
        
        public long nextLong() {
            return Long.parseLong(nextToken());
        }
        
        public double nextDouble() {
            return Double.parseDouble(nextToken());
        }
    }
}