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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import qualified Data.Map as M

select :: [a] -> [(a, [a])]
select [x]    = [(x, [])]
select (x:xs) = (x, xs) : map (\(y, ys) -> (y, x:ys)) (select xs)

permutations :: [a] -> [[a]]
permutations [] = [[]]
permutations xs =
  concatMap (\(y, ys) -> map (y:) (permutations ys)) $ select xs

main = do
 n <- read <$> getLine :: IO Int 
 let permutationsList = M.fromList $ zip (permutations [1..n]) [1..]
 pList <- map read . words <$> getLine :: IO[Int]
 qList <- map read . words <$> getLine :: IO[Int] 
 print $ abs $ (permutationsList M.! pList) - (permutationsList M.! qList)