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 Rust (1.15.1) (judge verdict: Accepted)

macro_rules! scan_sub {
    ( $itr:ident, $t:ty ) => {
        let mut buf = String::new();
        let $itr = {
            while buf.trim().len() == 0 {
                buf = String::new();
                std::io::stdin().read_line(&mut buf).unwrap();
            }
            buf.split_whitespace().map(|n| n.parse::<$t>().unwrap())
        };
    };
}

macro_rules! scan {
    ( $x:ident; Vec<$t:ty> ) => {
        scan_sub!(itr, $t);
        let $x = itr.collect::<Vec<_>>();
    };
    ( $( $x:ident ),*; $t:ty ) => {
        scan_sub!(itr, $t);
        let mut itr = itr;
        $(
            let $x = itr.next().unwrap();
        )*
    };
}

use std::collections::HashMap;

fn f(hm: &mut HashMap<u64, usize>, v: &Vec<usize>, n: u32, index: &mut usize, acc: u64) {
    if v.is_empty() {
        hm.insert(acc, *index);
        *index += 1;
        return;
    }

    for &t in v.iter() {
        let next_v = v.iter()
            .filter(|&&s| s != t)
            .map(|&s| s).collect::<Vec<_>>();
        f(hm, &next_v, n-1, index, acc+(t as u64 * 10_u64.pow(n-1)));
    }
}

fn main() {
    scan!(n; usize);
    scan!(p; Vec<u64>);
    scan!(q; Vec<u64>);

    let mut n_fact = 1;
    for i in 1..n+1 { n_fact *= i; }
    let mut dict = HashMap::with_capacity(n_fact);
    let v = (1..n+1).collect::<Vec<_>>();
    let mut index = 0;

    f(&mut dict, &v, n as u32, &mut index, 0);

    let mut p_num = 0;
    let mut q_num = 0;
    for i in 0..n {
        p_num += p[i] * 10_u64.pow((n-i-1) as u32);
        q_num += q[i] * 10_u64.pow((n-i-1) as u32);
    }

    let a = dict[&p_num];
    let b = dict[&q_num];

    let (a, b) = if a > b { (a, b) } else { (b, a) };
    println!("{}", a-b);
}