Problem:
Score : 300 points

Problem Statement

We have two permutations P and Q of size N (that is, P and Q are both rearrangements of (1,~2,~...,~N)).

There are N! possible permutations of size N. Among them, let P and Q be the a-th and b-th lexicographically smallest permutations, respectively. Find |a - b|.

Notes

For two sequences X and Y, X is said to be lexicographically smaller than Y if and only if there exists an integer k such that X_i = Y_i~(1 \leq i < k) and X_k < Y_k.

Constraints

2 \leq N \leq 8

P and Q are permutations of size N.

Input

Input is given from Standard Input in the following format:

N
P_1 P_2 ... P_N
Q_1 Q_2 ... Q_N

Output

Print |a - b|.

Sample Input 1

3
1 3 2
3 1 2

Sample Output 1

3

There are 6 permutations of size 3: (1,~2,~3), (1,~3,~2), (2,~1,~3), (2,~3,~1), (3,~1,~2), and (3,~2,~1). Among them, (1,~3,~2) and (3,~1,~2) come 2-nd and 5-th in lexicographical order, so the answer is |2 - 5| = 3.

Sample Input 2

8
7 3 5 4 2 1 6 8
3 8 2 5 4 6 7 1

Sample Output 2

17517

Sample Input 3

3
1 2 3
1 2 3

Sample Output 3

0

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Accepted)

(defun fact (n)
  (let ((fact 1))
    (loop :for i :from 2 :to n
          :do (setf fact (* i fact)))
    fact))

(defun permutaions (n)
  (let ((table (make-hash-table :size (fact n) :test 'equalp))
        (order 0))
    (labels ((next (n use vec)
               (if (= (length vec) n)
                   (setf (gethash (copy-seq vec) table) (incf order))
                   (loop :for i :from 1 :to n
                         :unless (aref use i)
                         :do (progn
                               (setf (aref use i) t)
                               (vector-push i vec)
                               (next n use vec)
                               (vector-pop vec)
                               (setf (aref use i) nil))))))
      (next n 
            (make-array `(,(1+ n)) :initial-element nil)
            (make-array `(,n) :adjustable t :fill-pointer 0))
      table)))

(let* ((n (read))
       (p (make-array `(,n)))
       (q (make-array `(,n)))
       (m (permutaions n)))
  (loop :for i :from 0 :to (1- n)
        :do (setf (aref p i) (read)))
  (loop :for i :from 0 :to (1- n)
        :do (setf (aref q i) (read)))
  (format t "~A~%" (abs (- (gethash p m) (gethash q m)))))