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.Linq;
using CompLib.Collections.Generic;
using CompLib.Util;

public class Program
{
    int N;
    int[] A;
    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        A = sc.IntArray();

        // iより左　最初A_iより大きいやつ
        var lF = new long[N];

        // iより左、2番目に大きいやつ
        var lS = new long[N];

        var rF = new long[N];
        var rS = new long[N];
        var st1 = new SegmentTree<long>(Math.Max, -1);
        var st2 = new SegmentTree<long>(Math.Min, N);
        for (int i = 0; i < N; i++)
        {
            lF[i] = st1.Query(A[i] + 1, N + 1);
            if (lF[i] != -1)
            {
                st1[A[lF[i]]] = -1;
                lS[i] = st1.Query(A[i] + 1, N + 1);
                st1[A[lF[i]]] = lF[i];
            }
            else
            {
                lS[i] = -1;
            }
            st1[A[i]] = i;

            int j = N - i - 1;
            rF[j] = st2.Query(A[j] + 1, N + 1);
            if (rF[j] != N)
            {
                st2[A[rF[j]]] = N;
                rS[j] = st2.Query(A[j] + 1, N + 1);
                st2[A[rF[j]]] = rF[j];
            }
            else
            {
                rS[j] = N;
            }
            st2[A[j]] = j;
        }

        long ans = 0;


        for (int i = 0; i < N; i++)
        {
            long l = (lF[i] - lS[i]) * (rF[i] - i);
            long r = (i - lF[i]) * (rS[i] - rF[i]);
            ans += (l + r) * A[i];
        }
        Console.WriteLine(ans);
    }

    public static void Main(string[] args) => new Program().Solve();
}

namespace CompLib.Collections.Generic
{
    using System;

    public class SegmentTree<T>
    {
        // 制約に合った2の冪
        private const int N = 1 << 21;
        private T[] _array;

        private T _identity;
        private Func<T, T, T> _operation;

        public SegmentTree(Func<T, T, T> operation, T identity)
        {
            _identity = identity;
            _operation = operation;
            _array = new T[N * 2];
            for (int i = 1; i < N * 2; i++)
            {
                _array[i] = _identity;
            }
        }

        /// <summary>
        /// A[i]をnに更新 O(log N)
        /// </summary>
        /// <param name="i"></param>
        /// <param name="n"></param>
        public void Update(int i, T n)
        {
            i += N;
            _array[i] = n;
            while (i > 1)
            {
                i /= 2;
                _array[i] = _operation(_array[i * 2], _array[i * 2 + 1]);
            }
        }

        private T Query(int left, int right, int k, int l, int r)
        {
            if (r <= left || right <= l)
            {
                return _identity;
            }

            if (left <= l && r <= right)
            {
                return _array[k];
            }

            return _operation(Query(left, right, k * 2, l, (l + r) / 2),
                Query(left, right, k * 2 + 1, (l + r) / 2, r));
        }

        /// <summary>
        /// A[left] op A[left+1] ... op A[right-1]を求める
        /// </summary>
        /// <param name="left"></param>
        /// <param name="right"></param>
        /// <returns></returns>
        public T Query(int left, int right)
        {
            return Query(left, right, 1, 0, N);
        }

        public T this[int i]
        {
            set { Update(i, value); }
            get { return _array[i + N]; }
        }
    }
}

namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}