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 bisect
N = int(input())
P = list(map(int, input().split()))
P_id = [0]*(N + 1)

for i in range(N):
    P_id[P[i]] = i

A = [- 1, P_id[N], N] 


res = 0
for i in range(N - 1, 0, - 1):
    j = bisect.bisect_left(A, P_id[i])
    A.insert(j, P_id[i])
    a = A[j]
    l1 = A[j - 1]
    r1 = A[j + 1]
    if l1 != - 1:
        l2 = A[j - 2]
    else:
        l2 = l1
    if r1 != N:
        r2 = A[j + 2]
    else:
        r2 = r1
    
    if l1 == - 1:
        res += i*(a + 1)*(r2 - r1)
    elif r1 == N:
        res += i*(l1 - l2)*(N - a)
    else:
        res += i*((a - l1)*(r2 - r1) + (l1 - l2)*(r1 - a))
print(res)