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.IO;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace AtCoder.Contest.E
{
	class Program
	{
		public static void Main(string[] args)
		{
			var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
			Console.SetOut(sw);

			Scanner cin = new Scanner();

			int n = cin.nextInt();
			var p = cin.ArrayInt(n);
			var s = new MultiSet<int>();
			var d = new Dictionary<int, int>();
			long ans = 0;
			for (int i = 0; i < n; i++)
			{
				d.Add(p[i], i + 1);
			}
			s.Insert(0);
			s.Insert(0);
			s.Insert(n + 1);
			s.Insert(n + 1);

			for (int i = n; i > 0; i--)
			{
				int l1, l2, r1, r2, x;
				l1 = s.LowerBound(d[i]) - 1;
				l2 = l1 - 1;
				r1 = s.UpperBound(d[i]);
				r2 = r1 + 1;
				x = d[i];
				long L1, L2, R1, R2;
				L1 = x - s[l1];
				L2 = s[l1] - s[l2];
				R1 = s[r1] - x;
				R2 = s[r2] - s[r1];
				ans += (L1 * R2 + L2 * R1) * i;
				s.Insert(d[i]);
			}
			Console.WriteLine(ans);

			Console.Out.Flush();
		}
	}

	/// <summary>
	/// C-like multiset
	/// </summary>
	public class MultiSet<T> : Set<T> where T : IComparable
	{
		public override void Insert(T v)
		{
			if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
			else _root = SB_BinarySearchTree<T>.Insert(_root, v);
		}
	}

	/// <summary>
	/// C-like set
	/// </summary>
	public class Set<T> where T : IComparable
	{
		protected SB_BinarySearchTree<T>.Node _root;

		public T this[int idx] { get { return ElementAt(idx); } }

		public int Count()
		{
			return SB_BinarySearchTree<T>.Count(_root);
		}

		public virtual void Insert(T v)
		{
			if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
			else
			{
				if (SB_BinarySearchTree<T>.Find(_root, v) != null) return;
				_root = SB_BinarySearchTree<T>.Insert(_root, v);
			}
		}

		public void Clear()
		{
			_root = null;
		}

		public void Remove(T v)
		{
			_root = SB_BinarySearchTree<T>.Remove(_root, v);
		}

		public bool Contains(T v)
		{
			return SB_BinarySearchTree<T>.Contains(_root, v);
		}

		public T ElementAt(int k)
		{
			var node = SB_BinarySearchTree<T>.FindByIndex(_root, k);
			if (node == null) throw new IndexOutOfRangeException();
			return node.Value;
		}

		public int Count(T v)
		{
			return SB_BinarySearchTree<T>.UpperBound(_root, v) - SB_BinarySearchTree<T>.LowerBound(_root, v);
		}

		public int LowerBound(T v)
		{
			return SB_BinarySearchTree<T>.LowerBound(_root, v);
		}

		public int UpperBound(T v)
		{
			return SB_BinarySearchTree<T>.UpperBound(_root, v);
		}

		public Tuple<int, int> EqualRange(T v)
		{
			if (!Contains(v)) return new Tuple<int, int>(-1, -1);
			return new Tuple<int, int>(SB_BinarySearchTree<T>.LowerBound(_root, v), SB_BinarySearchTree<T>.UpperBound(_root, v) - 1);
		}

		public List<T> ToList()
		{
			return new List<T>(SB_BinarySearchTree<T>.Enumerate(_root));
		}
	}

	/// <summary>
	/// Self-Balancing Binary Search Tree
	/// (using Randamized BST)
	/// </summary>
	public class SB_BinarySearchTree<T> where T : IComparable
	{
		public class Node
		{
			public T Value;
			public Node LChild;
			public Node RChild;
			public int Count;     //size of the sub tree

			public Node(T v)
			{
				Value = v;
				Count = 1;
			}
		}

		static Random _rnd = new Random();

		public static int Count(Node t)
		{
			return t == null ? 0 : t.Count;
		}

		static Node Update(Node t)
		{
			t.Count = Count(t.LChild) + Count(t.RChild) + 1;
			return t;
		}

		public static Node Merge(Node l, Node r)
		{
			if (l == null || r == null) return l == null ? r : l;

			if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble())
			{
				l.RChild = Merge(l.RChild, r);
				return Update(l);
			}
			else
			{
				r.LChild = Merge(l, r.LChild);
				return Update(r);
			}
		}

		/// <summary>
		/// split as [0, k), [k, n)
		/// </summary>
		public static Tuple<Node, Node> Split(Node t, int k)
		{
			if (t == null) return new Tuple<Node, Node>(null, null);
			if (k <= Count(t.LChild))
			{
				var s = Split(t.LChild, k);
				t.LChild = s.Item2;
				return new Tuple<Node, Node>(s.Item1, Update(t));
			}
			else
			{
				var s = Split(t.RChild, k - Count(t.LChild) - 1);
				t.RChild = s.Item1;
				return new Tuple<Node, Node>(Update(t), s.Item2);
			}
		}

		public static Node Remove(Node t, T v)
		{
			if (Find(t, v) == null) return t;
			return RemoveAt(t, LowerBound(t, v));
		}

		public static Node RemoveAt(Node t, int k)
		{
			var s = Split(t, k);
			var s2 = Split(s.Item2, 1);
			return Merge(s.Item1, s2.Item2);
		}

		public static bool Contains(Node t, T v)
		{
			return Find(t, v) != null;
		}

		public static Node Find(Node t, T v)
		{
			while (t != null)
			{
				var cmp = t.Value.CompareTo(v);
				if (cmp > 0) t = t.LChild;
				else if (cmp < 0) t = t.RChild;
				else break;
			}
			return t;
		}

		public static Node FindByIndex(Node t, int idx)
		{
			if (t == null) return null;

			var currentIdx = Count(t) - Count(t.RChild) - 1;
			while (t != null)
			{
				if (currentIdx == idx) return t;
				if (currentIdx > idx)
				{
					t = t.LChild;
					currentIdx -= (Count(t == null ? null : t.RChild) + 1);
				}
				else
				{
					t = t.RChild;
					currentIdx += (Count(t == null ? null : t.LChild) + 1);
				}
			}

			return null;
		}

		public static int UpperBound(Node t, T v)
		{
			var torg = t;
			if (t == null) return -1;

			var ret = Int32.MaxValue;
			var idx = Count(t) - Count(t.RChild) - 1;
			while (t != null)
			{
				var cmp = t.Value.CompareTo(v);

				if (cmp > 0)
				{
					ret = Math.Min(ret, idx);
					t = t.LChild;
					idx -= (Count(t == null ? null : t.RChild) + 1);
				}
				else if (cmp <= 0)
				{
					t = t.RChild;
					idx += (Count(t == null ? null : t.LChild) + 1);
				}
			}
			return ret == Int32.MaxValue ? Count(torg) : ret;
		}

		public static int LowerBound(Node t, T v)
		{
			var torg = t;
			if (t == null) return -1;

			var idx = Count(t) - Count(t.RChild) - 1;
			var ret = Int32.MaxValue;
			while (t != null)
			{
				var cmp = t.Value.CompareTo(v);
				if (cmp >= 0)
				{
					if (cmp == 0) ret = Math.Min(ret, idx);
					t = t.LChild;
					if (t == null) ret = Math.Min(ret, idx);
					idx -= t == null ? 0 : (Count(t.RChild) + 1);
				}
				else if (cmp < 0)
				{
					t = t.RChild;
					idx += (Count(t == null ? null : t.LChild) + 1);
					if (t == null) return idx;
				}
			}
			return ret == Int32.MaxValue ? Count(torg) : ret;
		}

		public static Node Insert(Node t, T v)
		{
			var ub = LowerBound(t, v);
			return InsertByIdx(t, ub, v);
		}

		static Node InsertByIdx(Node t, int k, T v)
		{
			var s = Split(t, k);
			return Merge(Merge(s.Item1, new Node(v)), s.Item2);
		}

		public static IEnumerable<T> Enumerate(Node t)
		{
			var ret = new List<T>();
			Enumerate(t, ret);
			return ret;
		}

		static void Enumerate(Node t, List<T> ret)
		{
			if (t == null) return;
			Enumerate(t.LChild, ret);
			ret.Add(t.Value);
			Enumerate(t.RChild, ret);
		}
	}

	class Scanner
	{
		string[] s;
		int i;

		char[] cs = new char[] { ' ' };

		public Scanner()
		{
			s = new string[0];
			i = 0;
		}

		public string next()
		{
			if (i < s.Length) return s[i++];
			string st = Console.ReadLine();
			while (st == "") st = Console.ReadLine();
			s = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
			if (s.Length == 0) return next();
			i = 0;
			return s[i++];
		}

		public int nextInt()
		{
			return int.Parse(next());
		}

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = nextInt() + add;
			}
			return Array;
		}

		public long nextLong()
		{
			return long.Parse(next());
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = nextLong() + add;
			}
			return Array;
		}

		public double nextDouble()
		{
			return double.Parse(next());
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = nextDouble() + add;
			}
			return Array;
		}
	}
}