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 OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

module M = Map.Make (struct type t = int let compare (a:int) (b:int) = compare a b end)

let () =
    let split s =
        let rec loop i cur flag acc =
            if i = String.length s then ( if flag then cur :: acc else acc) else ( match flag, s.[i] with
                | true,  ' ' -> loop (i + 1) "" false (cur :: acc)
                | false, ' ' -> loop (i + 1) "" false acc
                | true,  v   -> loop (i + 1) (Printf.sprintf "%s%c" cur v) true acc
                | false, v   -> loop (i + 1) (Printf.sprintf "%s%c" cur v) true acc
            )
        in
        List.map int_of_string (List.rev (loop 0 "" false []))
    in
    let m = 1 lsl 20 in
    let main () =
        let n = int_of_string (read_line ()) in
        let p = Array.of_list (split (read_line ())) in
        let rec loop i map acc =
            if i = n then acc else
                let next = M.fold (fun org v next ->
                    let first = org / m in
                    let second = org mod m in
                    let key = if p.(i) > first then p.(i) * m + first else
                        if p.(i) > second then first * m + p.(i) else org
                    in
                    M.add key (v + try M.find key next with _ -> 0) next) map M.empty
                in
                let acc = acc + M.fold (fun org v acc -> (org mod m) * v + acc) next 0 in
                loop (i + 1) (M.add (p.(i) * m) 1 next) acc
        in
        let r = loop 1 (M.singleton (p.(0) * m) 1) 0 in
        Printf.printf "%d\n" r
    in
    main ()