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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

import heapq

n = int(input())
p = list(map(int, input().split()))

dic = [0] * n
for i in range(n):
    dic[p[i]-1] = i+1

ans = 0
big = {0,n+1}
big.add(dic[n-1])
for i in range(n-1,0,-1):
    i_no = dic[i-1]
    bigger = big & set(range(i_no+1,n+2,1))
    smaller = big & set(range(i_no-1,-1,-1))

    if len(bigger)==1:
        r1 = n+1
        r2 = n+1
    else:
        r1, r2 = heapq.nsmallest(2, bigger)
        # r1 = min(bigger)
        # bigger.discard(r1)
        # r2 = min(bigger)

    if len(smaller)==1:
        l1 = 0
        l2 = 0
    else:
        l1, l2 = heapq.nlargest(2, smaller)
        # l1 = max(smaller)
        # smaller.discard(l1)
        # l2 = max(smaller)

    ans += i*((r2-r1)*(i_no-l1)+(r1-i_no)*(l1-l2))

    big.add(i_no)

print(ans)