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;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using BitArray = System.Collections.BitArray;
using BigInteger = System.Numerics.BigInteger;
using TextReader = System.IO.TextReader;
using static AtCoderProject.Global;

namespace AtCoderProject
{
    public class Program
    {
        public object Calc()
        {
            var N = consoleReader.Int;
            var p = consoleReader.Split.Int0;
            var pIndex = new int[N];
            for (int i = 0; i < p.Length; i++)
                pIndex[p[i]] = i;

            var lefts = new int[N];
            var rights = new int[N];
            var lefts2 = new int[N];
            var rights2 = new int[N];
            var indexes = new SortedCollection<int>(N);
            indexes.Add(pIndex[pIndex.Length - 1]);
            lefts[pIndex[pIndex.Length - 1]] = -1;
            lefts2[pIndex[pIndex.Length - 1]] = -1;
            rights[pIndex[pIndex.Length - 1]] = N;
            rights2[pIndex[pIndex.Length - 1]] = N;
            for (int i = p.Length - 2; i >= 0; i--)
            {
                var m = indexes.Add(pIndex[i]);
                if (m == indexes.Count - 1)
                {
                    rights[pIndex[i]] = N;
                    rights2[pIndex[i]] = N;
                }
                else if (m == indexes.Count - 2)
                {
                    rights[pIndex[i]] = indexes[m + 1];
                    rights2[pIndex[i]] = N;
                }
                else
                {
                    rights[pIndex[i]] = indexes[m + 1];
                    rights2[pIndex[i]] = indexes[m + 2];
                }


                if (m == 0)
                {
                    lefts[pIndex[i]] = -1;
                    lefts2[pIndex[i]] = -1;
                }
                else if (m == 1)
                {
                    lefts[pIndex[i]] = indexes[m - 1];
                    lefts2[pIndex[i]] = -1;
                }
                else
                {
                    lefts[pIndex[i]] = indexes[m - 1];
                    lefts2[pIndex[i]] = indexes[m - 2];
                }
            }

            var res = new long[N];
            for (int i = 0; i < N; i++)
            {
                res[i] = (lefts[i] - lefts2[i]) * (rights[i] - i)
                    + (i - lefts[i]) * (rights2[i] - rights[i]);
                res[i] *= p[i] + 1;
            }

            return res.Sum();
        }
        class SortedCollection<T> : IList<T>
        {
            List<T> list;

            public SortedCollection() : this(Comparer<T>.Default) { }
            public SortedCollection(int capacity) : this(capacity, Comparer<T>.Default) { }
            public SortedCollection(IComparer<T> comparer) { Comparer = comparer; list = new List<T>(); }
            public SortedCollection(int capacity, IComparer<T> comparer) { Comparer = comparer; list = new List<T>(capacity); }

            public IComparer<T> Comparer { get; }
            public bool IsReadOnly => false;
            public int Count => list.Count;

            public T this[int index]
            {
                get { return list[index]; }
                set { list[index] = value; }
            }
            public int BinarySearch(T item) => ~list.BinarySearch(item, Comparer);

            public int Add(T item)
            {
                var index = BinarySearch(item);
                list.Insert(index, item);
                return index;
            }
            void ICollection<T>.Add(T item) => this.Add(item);

            public void Clear() => list.Clear();
            public bool Contains(T item) => list.Count > BinarySearch(item);
            public void CopyTo(T[] array, int arrayIndex) => list.CopyTo(array, arrayIndex);

            public bool Remove(T item)
            {
                int index = this.IndexOf(item);
                if (index < 0)
                    return false;
                list.RemoveAt(index);
                return true;
            }

            IEnumerator IEnumerable.GetEnumerator() => this.GetEnumerator();
            public IEnumerator<T> GetEnumerator() => list.GetEnumerator();

            public int IndexOf(T item)
            {
                var index = BinarySearch(item);
                if (index < list.Count) return index;
                else return -1;
            }

            public void Insert(int index, T item)
            {
                throw new NotImplementedException();
            }

            public void RemoveAt(int index) => list.RemoveAt(index);

        }

        #region いつもの
#pragma warning disable
        [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] private ConsoleReader consoleReader;
        public Program(ConsoleReader consoleReader) { this.consoleReader = consoleReader; }
        static void Main() => Console.WriteLine(new Program(new ConsoleReader(Console.In)).Calc()); static string AllLines<T>(IEnumerable<T> source) => string.Join("\n", source);
    }
    static class Global
    {
        private class ComparerReverseImpl<T> : Comparer<T> where T : IComparable<T> { public override int Compare(T y, T x) => x.CompareTo(y); public override bool Equals(object obj) => obj != null && GetType() == obj.GetType(); public override int GetHashCode() => GetType().GetHashCode(); }
        public static IComparer<T> ComparerReverse<T>() where T : IComparable<T> => new ComparerReverseImpl<T>();
        public static BigInteger ParseBigInteger(string s)
        {
            // MonoのBigInteger.Parseが遅いので自前実装
            var res = BigInteger.Zero;
            var splited = new string[(s.Length + 7) / 8];
            for (int i = 0; i < splited.Length - 1; i++)
            {
                splited[i] = s.Substring(8 * i, 8);
            }
            splited[splited.Length - 1] = s.Substring(8 * (splited.Length - 1));
            foreach (var sp in splited)
            {
                res *= (int)Math.Pow(10, sp.Length);
                res += int.Parse(sp);
            }
            return res;
        }
    }
    static class Ext
    {
        public static IComparer<T> Reverse<T>(this IComparer<T> comparer) => Comparer<T>.Create((x, y) => comparer.Compare(y, x));
        public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count());
        public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i);
    }
    public class ConsoleReader { private string[] ReadLineSplit() => textReader.ReadLine().Split(Array.Empty<char>(), StringSplitOptions.RemoveEmptyEntries); private string[] line = Array.Empty<string>(); private int linePosition; private TextReader textReader; public ConsoleReader(TextReader tr) { textReader = tr; } public int Int => int.Parse(String); public int Int0 => Int - 1; public long Long => long.Parse(String); public double Double => double.Parse(String); public string String { get { if (linePosition >= line.Length) { linePosition = 0; line = ReadLineSplit(); } return line[linePosition++]; } } public class SplitLine { private string[] splited; public SplitLine(ConsoleReader cr) { splited = cr.ReadLineSplit(); cr.line = Array.Empty<string>(); } public int[] Int => String.Select(x => int.Parse(x)).ToArray(); public int[] Int0 => String.Select(x => int.Parse(x) - 1).ToArray(); public long[] Long => String.Select(x => long.Parse(x)).ToArray(); public double[] Double => String.Select(x => double.Parse(x)).ToArray(); public string[] String => splited; } public SplitLine Split => new SplitLine(this); public class RepeatReader : IEnumerable<ConsoleReader> { ConsoleReader cr; int count; public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; } public IEnumerator<ConsoleReader> GetEnumerator() => Enumerable.Repeat(cr, count).GetEnumerator(); System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator(); public IEnumerable<string> String => this.Select(cr => cr.String); public IEnumerable<int> Int => this.Select(cr => cr.Int); public IEnumerable<int> Int0 => this.Select(cr => cr.Int - 1); public IEnumerable<long> Long => this.Select(cr => cr.Long); public IEnumerable<double> Double => this.Select(cr => cr.Double); } public RepeatReader Repeat(int count) => new RepeatReader(this, count); }
    #endregion
}