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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil nil t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

;;;
;;; Disjoint sparse table on arbitrary semigroup
;;;

;;; Reference:
;;; https://discuss.codechef.com/questions/117696/tutorial-disjoint-sparse-table
;;; http://noshi91.hatenablog.com/entry/2018/05/08/183946 (Japanese)
;;; http://drken1215.hatenablog.com/entry/2018/09/08/162600 (Japanese)

;; NOTE: This constructor is slow on SBCL version earlier than 1.5.6 as the type
;; propagation of MAKE-ARRAY doesn't work. The following files are required to
;; enable the optimization.
;; version < 1.5.0: array-element-type.lisp, make-array-header.lisp
;; version < 1.5.6: make-array-header.lisp
(declaim (inline make-disjoint-sparse-table))
(defun make-disjoint-sparse-table (vector binop)
  "BINOP := binary operator (comprising a semigroup)"
  (let* ((n (length vector))
         (height (integer-length (- n 1)))
         (table (make-array (list height n) :element-type 'uint32)))
    (dotimes (j n)
      (setf (aref table 0 j) (aref vector j)))
    (do ((i 1 (+ i 1)))
        ((>= i height))
      (let* ((width/2 (ash 1 i))
             (width (* width/2 2)))
        (do ((j 0 (+ j width)))
            ((>= j n))
          (let ((mid (min (+ j width/2) n)))
            ;; fill the first half
            (setf (aref table i (- mid 1))
                  (aref vector (- mid 1)))
            (do ((k (- mid 2) (- k 1)))
                ((< k j))
              (setf (aref table i k)
                    (funcall binop (aref vector k) (aref table i (+ k 1)))))
            (when (>= mid n)
              (return))
            ;; fill the second half
            (setf (aref table i mid)
                  (aref vector mid))
            (let ((end (min n (+ mid width/2))))
              (do ((k (+ mid 1) (+ k 1)))
                  ((>= k end))
                (setf (aref table i k)
                      (funcall binop (aref table i (- k 1)) (aref vector k)))))))))
    table))

(declaim (inline dst-query))
(defun dst-query (table binop left right)
  "Queries the interval [LEFT, RIGHT)."
  (declare ((integer 0 #.most-positive-fixnum) left right)
           ((simple-array * (* *)) table))
  (when (= left right)
    (return-from dst-query 0))
  (setq right (- right 1)) ;; change to closed interval
  (if (= left right)
      (aref table 0 left)
      (let ((h (- (integer-length (logxor left right)) 1)))
        (funcall binop
                 (aref table h left)
                 (aref table h right)))))

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (declare #.OPT)
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10)) result))))
              (return (if minus (- result) result))))))))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(defun main ()
  (declare #.OPT)
  (let* ((n (read))
         (ps (make-array 100000 :element-type 'uint32)))
    (declare (uint31 n))
    (dotimes (i n)
      (setf (aref ps i) (read-fixnum)))
    (let ((dtable (make-disjoint-sparse-table ps #'max)))
      (let ((res 0))
        (declare (fixnum res))
        (dotimes (i n)
          (let ((p (aref ps i)))
            (declare (uint31 p))
            (let* ((l (sb-int:named-let bisect ((ok -1) (ng i))
                        (declare (int32 ok ng))
                        (if (<= (- ng ok) 1)
                            ok
                            (let* ((mid (ash (+ ok ng) -1))
                                   (val (dst-query dtable #'max mid i)))
                              (declare (uint32 val))
                              (if (> val p)
                                  (bisect mid ng)
                                  (bisect ok mid))))))
                   (r+1 (sb-int:named-let bisect ((ng i) (ok (+ n 1)))
                          (declare (int32 ng ok))
                          (if (<= (- ok ng) 1)
                              ok
                              (let* ((mid (ash (+ ok ng) -1))
                                     (val (dst-query dtable #'max i mid)))
                                (declare (uint32 val))
                                (if (> val p)
                                    (bisect ng mid)
                                    (bisect mid ok))))))
                   (r (- r+1 1)))
              (when (>= l 0)
                (let* ((ll (sb-int:named-let bisect ((ok -1) (ng l))
                             (declare (int32 ok ng))
                             (if (<= (- ng ok) 1)
                                 ok
                                 (let* ((mid (ash (+ ok ng) -1))
                                        (val (dst-query dtable #'max mid l)))
                                   (declare (uint32 val))
                                   (if (> val p)
                                       (bisect mid ng)
                                       (bisect ok mid)))))))
                  (incf res (* (- r i)
                               (- l ll)
                               p))))
              (when (< r n)
                (let* ((rr+1 (sb-int:named-let bisect ((ng (+ r 1)) (ok (+ n 1)))
                               (declare (int32 ng ok))
                               (if (<= (- ok ng) 1)
                                   ok
                                   (let* ((mid (ash (+ ok ng) -1))
                                          (val (dst-query dtable #'max r+1 mid)))
                                     (declare (uint32 val))
                                     (if (> val p)
                                         (bisect ng mid)
                                         (bisect mid ok))))))
                       (rr (- rr+1 1)))
                  (incf res (* (- i l)
                               (- rr r)
                               p)))))))
        (println res)))))

#-swank (main)