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.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
using System.Reflection;
using static System.Math;
using System.Numerics;
static class Program{
	const int mod=(int)1e9+7;
	static void Main(){
		Sc sc=new Sc();
		var n=sc.I;
		var a=sc.Ia;
		long ans=0;
		St stmax=new St(n+2,true);
		St stmin=new St(n+2,false);
		stmax.Ud(0,0);
		stmax.Ud(n+1,0);
		stmin.Ud(0,n+1);
		stmin.Ud(n+1,n+1);
		int[] h=new int[n+1];
		for(int i = 0;i<n;i++) {h[a[i]]=i+1;}
		for(int i = n;i>0;i--) {
			int l=stmax.Get(0,h[i]-1).n,r=stmin.Get(h[i]+1,n+1).n;
			ans+=i*((long)l-stmax.Get(0,Max(l-1,0)).n)*(r-h[i]);
			ans+=i*((long)stmin.Get(Min(r+1,n+1),n+1).n-r)*(h[i]-l);
			stmax.Ud(h[i],h[i]);
			stmin.Ud(h[i],h[i]);
		}
		Console.WriteLine("{0}",ans);
	}
}
public class St{
	public class Mo{
		public int n;
		public long d;
		public Mo(long d,int n){this.d=d;this.n=n;}
	}
	public Mo[] dat;
	private int m=1;
	private bool mm;
	private Mo du;
	readonly private Func<Mo,Mo,bool> compare;
	public St(int n,bool mm){
		this.mm=mm;
		if(mm){du=new Mo(long.MinValue,-1);compare=CoMax;}
		else{du=new Mo(long.MaxValue,-1);compare=CoMin;}
		while(m<n){m<<=1;}
		dat=new Mo[m+n];
		for(int i=0;i<m+n;i++){dat[i]=du;}
	}
	private bool CoMax(Mo x,Mo y){return x.d>y.d;}
	private bool CoMin(Mo x,Mo y){return x.d<y.d;}
	public void Ud(int q,long b){
		q+=m-1;
		if(dat[q]==du){
			int z=q;
			dat[q]=new Mo(b,q-(m-1));
			while(q>0&&compare(dat[z],dat[(q-1)>>1])){
				dat[(q-1)>>1]=dat[z];
				q=(q-1)>>1;
			}
		}
		else{
			b+=dat[q].d;
			dat[q].d=du.d;
			dat[q]=new Mo(b,q-(m-1));
			q=(q-1)>>1;
			while(q>=0){
				Mo o=compare(dat[2*q+1],dat[2*q+2])?dat[2*q+1]:dat[2*q+2];
				if(dat[q]==o){break;}
				dat[q]=o;
				q=(q-1)>>1;
			}
		}
	}
	private Mo Fdg(int a,int b,int k,int l,int r){
		if(r<a||b<l){return du;}
		if(a<=l&&r<=b){return dat[k];}
		Mo p=Fdg(a,b,k*2+1,l,(l+r)>>1);
		Mo q=Fdg(a,b,k*2+2,(l+r+1)>>1,r);
		return compare(p,q)?p:q;
	}
	public Mo Get(int a,int b){return Fdg(a,b,0,0,m-1);}
}

public class Sc{
	public int I{get{return int.Parse(Console.ReadLine());}}
	public long L{get{return long.Parse(Console.ReadLine());}}
	public double D{get{return double.Parse(Console.ReadLine());}}
	public string S{get{return Console.ReadLine();}}
	public int[] Ia{get{return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public object[] Oa{get{return Console.ReadLine().Split();}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public int[] Ia3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),int.Parse);}
	public int[] Ia3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),int.Parse);}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public long[] La3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),long.Parse);}
	public long[] La3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),long.Parse);}
	public T[] Arr<T>(int n,Func<T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f();}return a;}
	public T[] Arr<T>(int n,Func<int,T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i);}return a;}
	public T[] Arr<T>(int n,Func<string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(Console.ReadLine().Split());}return a;}
	public T[] Arr<T>(int n,Func<int,string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i,Console.ReadLine().Split());}return a;}
}