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 Ruby (2.7.1) (judge verdict: Accepted)

n=gets.to_i

sam1=Array.new(n)
sam2=Array.new(n)
sam1=gets.split.map!(&:to_i)
sam2=gets.split.map!(&:to_i)

kot1=1
kot2=1
kazu1=n
kazu2=n
for i in 0..n-2
  ans1=1
  for j in 1..kazu1
    ans1=ans1*j
  end
  kot1+=ans1/kazu1*(sam1[i]-1)
  kazu1-=1
  for k in 0..n-1
    if sam1[k]>sam1[i]
      sam1[k]=sam1[k]-1
    end
  end
end
for i in 0..n-2
  ans2=1
  for j in 1..kazu2
    ans2=ans2*j
  end
  kot2+=ans2/kazu2*(sam2[i]-1)
  kazu2-=1
  for j in 0..n-1
    if sam2[j]>sam2[i]
      sam2[j]-=1
    end
  end
end
puts (kot1-kot2).abs