G - Three Permutations

Score : $600$ points

Problem Statement

Given are permutations of $(1, \dots, N)$: $p = (p_1, \dots, p_N)$ and $q = (q_1, \dots, q_N)$.

Find the number, modulo $(10^9 + 7)$, of permutations $r = (r_1, \dots, r_N)$ of $(1, \dots, N)$ such that $r_i \neq p_i$ and $r_i \neq q_i$ for every $i$ $(1 \leq i \leq N)$.

Constraints

• $1 \leq N \leq 3000$
• $1 \leq p_i, q_i \leq N$
• $p_i \neq p_j \, (i \neq j)$
• $q_i \neq q_j \, (i \neq j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$p_1$ $\ldots$ $p_N$

$q_1$ $\ldots$ $q_N$

Output

Print the answer.

4
1 2 3 4
2 1 4 3

4

There are four valid permutations: $(3, 4, 1, 2)$, $(3, 4, 2, 1)$, $(4, 3, 1, 2)$, and $(4, 3, 2, 1)$.

3
1 2 3
2 1 3

0

The answer may be $0$.

20
2 3 15 19 10 7 5 6 14 13 20 4 18 9 17 8 12 11 16 1
8 12 4 13 19 3 10 16 11 9 1 2 17 6 5 18 7 14 20 15

803776944

Be sure to print the count modulo $(10^9 + 7)$.