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 Fortran (gfortran v4.8.4) (judge verdict: Wrong Answer)

program main

  implicit none

  integer :: a, N, i, j, k, pn, qn, tmp
  integer,allocatable :: P(:), Q(:)
  
  read(*,*) N
  allocate( P(N) )
  allocate( Q(N) )
  read(*,*) P
  read(*,*) Q
  
  pn = 0 
  qn = 0 

  do i = 1, N
    ! kaijou
    tmp = 1
    do j = 1, N - i
      tmp =  tmp * j  
    end do
    a = 0
    do k = 1, i 
      if ( P(k) < P(i) ) a = a + 1
    end do
    pn = pn + ( P(i) - a - 1 ) * tmp
  end do
    pn = pn + 1 

  do i = 1, N 
    ! kaijou
    tmp = 1
    do j = 1, N - i
      tmp =  tmp * j  
    end do
    a = 0
    do k = 1, i
      if ( Q(k) < Q(i) ) a = a + 1
    end do
    qn = qn + ( Q(i) - a - 1 ) * tmp
  end do
    qn = qn + 1

  tmp = pn - qn  
  print*, abs( tmp ), pn, qn
  
end program main