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 Kotlin (1.0.0) (judge verdict: Wrong Answer)

import java.util.*

fun simple(P: List<Long>): Long {
    var res = 0L
    for (i in P.indices) {
        for (k in 2..P.size - i) {
            val l = P.subList(i, i + k).sorted()
            res += l[l.size - 2]
        }
    }
    return res
}

fun main(args: Array<String>) {
    val N = readLine()!!.toInt()
    val P = readLine()!!.split(" ").map{it.toLong()}

    fun calc(p: List<Long>): Pair<ArrayList<Long>, ArrayList<Long>> {
        val resMx = ArrayList<Long>()
        val res2nd = ArrayList<Long>()
        val Q = ArrayDeque<Pair<Long, Int>>()
        val mp = TreeMap<Long, Int>()

        for (i in p.indices) {
            val R = ArrayDeque<Pair<Long, Int>>()
            while (Q.isNotEmpty() && Q.last.first <= p[i]) {
                R.addLast(Q.pollLast())
            }

            while(R.isNotEmpty() && mp.isNotEmpty() && mp.firstKey() <= R.last.second) {
                mp.pollFirstEntry()
            }
            for (e in R) {
                mp[e.first] = e.second
            }

            if (Q.isEmpty()) {
                resMx.add(i + 1L)
                res2nd.add(i + 1L)
            } else {
                resMx.add(i - Q.last.second.toLong())

                var t = i + 1L
                if (mp.isNotEmpty() && mp.lastKey() >= p[i]) {
                    t = Math.min(t, i - mp.ceilingEntry(p[i]).value.toLong())
                }
                if (Q.size > 1) {
                    val r = Q.pollLast()
                    t = Math.min(t, i - Q.last.second.toLong())
                    Q.addLast(r)
                }
                res2nd.add(t)
            }

            Q.addLast(Pair(p[i], i))
        }

        return Pair(resMx, res2nd)
    }

    val (lMx, l2nd) = calc(P)
    val (rMx, r2nd) = calc(P.reversed())
    rMx.reverse()
    r2nd.reverse()

    var res = 0L
    for (i in P.indices) {
        res += lMx[i] * (r2nd[i] - rMx[i]) * P[i]
        res += (l2nd[i] - lMx[i]) * rMx[i] * P[i]
    }

    println(res)
//    println(simple(P))
}