Problem:
Score: 500 points

Problem Statement

Given is a permutation P of \{1, 2, \ldots, N\}.

For a pair (L, R) (1 \le L \lt R \le N), let X_{L, R} be the second largest value among P_L, P_{L+1}, \ldots, P_R.

Find \displaystyle \sum_{L=1}^{N-1} \sum_{R=L+1}^{N} X_{L,R}.

Constraints

2 \le N \le 10^5

1 \le P_i \le N

P_i \neq P_j (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
P_1 P_2 \ldots P_N

Output

Print \displaystyle \sum_{L=1}^{N-1} \sum_{R=L+1}^{N} X_{L,R}.

Sample Input 1

3
2 3 1

Sample Output 1

5

X_{1, 2} = 2, X_{1, 3} = 2, and X_{2, 3} = 1, so the sum is 2 + 2 + 1 = 5.

Sample Input 2

5
1 2 3 4 5

Sample Output 2

30

Sample Input 3

8
8 2 7 3 4 5 6 1

Sample Output 3

136

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;

class E
{
	static void Main()
	{
		var n = int.Parse(Console.ReadLine());
		var p = Console.ReadLine().Split().Select(int.Parse).ToArray();
		var q = new int[n + 1];
		for (int i = 0; i < n; i++) q[p[i]] = i;

		var bigs = new List<int> { q[n] };
		var s = 0L;
		for (int i = n - 1; i > 0; i--)
		{
			var bi = ~bigs.BinarySearch(q[i]);
			if (bi > 0)
			{
				var li = bigs[bi - 1];
				var li2 = bi - 2 >= 0 ? bigs[bi - 2] : -1;
				var ri2 = bi < bigs.Count ? bigs[bi] : n;
				s += (long)i * (li - li2) * (ri2 - q[i]);
			}
			if (bi < bigs.Count)
			{
				var ri = bigs[bi];
				var ri2 = bi + 1 < bigs.Count ? bigs[bi + 1] : n;
				var li2 = bi - 1 >= 0 ? bigs[bi - 1] : -1;
				s += (long)i * (q[i] - li2) * (ri2 - ri);
			}
			bigs.Insert(bi, q[i]);
		}
		Console.WriteLine(s);
	}
}