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 7.10.3) (judge verdict: Accepted)

{-# LANGUAGE FlexibleInstances #-}

import Data.List

-- stdin
class In a where stdin :: IO a
instance In [Int] where stdin = map read . words <$> getLine

type N = Int
type ChunkCount = Int

getIndex :: [Int] -> Int
getIndex ns = treeBottom (search ns []) [1..(product ns)]
  where
    -- [3, 2, 1, 4] []                       -> [(2, 4)]
    -- [2, 1, 4]    [(2, 4)]                 -> [(2, 4), (1, 3)]
    -- [1, 4]       [(2, 4), (1, 3)]         -> [(2, 4), (1, 3), (0, 2)]
    -- [4]          [(2, 4), (1, 3), (0, 2)] -> [(2, 4), (1, 3), (0, 2), (0, 1)]
    search :: [Int] -> [(N, ChunkCount)] -> [(N, ChunkCount)]
    search [] acc = acc
    search all@(n:ns) acc = search ns $ acc ++ [(length $ takeWhile (/=n) $ sort all, length all)]

    -- [(2, 4), (1, 3), (0, 2), (0, 1)] [1..24] -> 15
    treeBottom :: [(N, ChunkCount)] -> [Int] -> Int
    treeBottom [] [n] = n
    treeBottom ((n,cc):remains) ns = treeBottom remains $ chunk ns cc !! n
      where
        -- [1, 2, 3, 4, 5, 6] -> [[1, 2], [3, 4], [5, 6]]
        chunk :: [Int] -> ChunkCount -> [[Int]]
        chunk ns n = chunk_ ns (length ns `div` n) []
          where
            chunk_ [] _ acc = acc
            chunk_ ns len acc = chunk_ (drop len ns) len (acc ++ [take len ns])

main = do
    _ <- getLine
    a <- stdin
    b <- stdin

    print $ abs (getIndex a - getIndex b)