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 PyPy3 (2.4.0) (judge verdict: Time Limit Exceeded)

def main():
    import sys
    input = sys.stdin.readline
    sys.setrecursionlimit(10**7)
    from collections import Counter, deque
    from itertools import combinations, permutations, accumulate, groupby, product
    from bisect import bisect_left,bisect_right
    from heapq import heapify, heappop, heappush
    import math
    #from math import gcd

    #inf = 10**17
    #mod = 10**9 + 7

    n = int(input())
    p = list(map(int, input().split()))
    l = []
    for i in range(n):
        l.append([p[i], i])
    l.sort(reverse=True)
    s = [-1, n]
    res = 0
    
    for num, idx in l:
        if len(s) == 2:
            s = [s[0], idx, s[1]]
        else:
            a = bisect_left(s, idx)
            if a == 1:
                res += num * ((s[a+1]-s[a]) * (idx+1))
            elif a== len(s)-1:
                res += num * ((n-idx) * (s[a-1]-s[a-2]))
            else:
                res += num * ((s[a+1]-s[a]) * (idx-s[a-1]) + (s[a]-idx) * (s[a-1]-s[a-2]))
            s.insert(a, idx)
    print(res)

if __name__ == '__main__':
    main()