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: Runtime Error)

n = int(input())
p = [int(i) for i in input().split()]
ans = 0
for i in range(n): #左端から
  if i==0: #一番左なら
    if p[i]==n:
      ans += 0
    elif p[i]==n-1:
      idx = p.index(n)
      ans += p[i]*(n-idx)
    else:
      idx = []
      for j in reversed(range(p[i]+1,n+1)):
        idx.append(p.index(j))
      idx.sort()
      temp = idx[1]-idx[0]
      ans += p[i]*temp
  elif i==n-1: #右端なら
    if p[i]==n:
      ans += 0
    elif p[i]==n-1:
      idx = p.index(n)
      ans += p[i]*(idx+1)
    else:
      idx = []
      for j in reversed(range(p[i]+1,n+1)):
        idx.append(p.index(j))
      idx.sort()
      temp = idx[-1] - idx[-2]
      ans += p[i]*temp
  else: #真ん中なら
    if p[i]==n:
      ans+=0
    elif p[i]==n-1:
      idx = p.index(n)
      chi = min(i,idx)
      ook = max(i,idx)
      temp = (chi+1)*(n-ook)
      ans += p[i]*temp
    else: #「大」が2つ以上あるとき
      left = p[:i]
      right = p[i+1:]
      if max(left)<p[i]: #小●大
        idx = []
        for j in range(i+1,n):
          if p[j]>p[i] and (len(idx)<2):
            idx.append(j)
          elif len(idx)==2:
            break
          else:
            idx = idx
        temp = (i+1)*(idx[1]-idx[0])
        ans += p[i]*temp
      elif max(right)>p[i]: #大●小
        idx = []
        for j in reversed(range(i)):
          if p[j]>p[i] and (len(idx)<2):
            idx.append(j)
          elif len(idx)==2:
            break
          else:
            idx = idx
        temp = (n-i)*(idx[0]-idx[1])
        ans += temp*p[i]
      else:
        idx_left = []
        for k in reversed(range(i)):
          if p[k]>p[i] and (len(idx_left)<2):
            idx_left.append(k)
          elif len(idx_left)==2:
            break
          else:
            idx_left = idx_left
        if len(idx_left)<2:
          idx_left.append(-1)
        idx_right = []
        for l in range(i+1,n):
          if p[l]>p[i] and (len(idx_right)<2):
            idx_right.append(l)
          elif len(idx_right)==2:
            break
          else:
            idx_right = idx_right
        if len(idx_right)<2:
          idx_right.append(n)
        temp_left = (idx_right[0]-i)*(idx_left[0]-idx_left[1])
        temp_right = (idx_right[1]-idx_right[0])*(i-idx_left[0])
        temp = temp_left+temp_right
        ans += p[i]*temp
print(ans)