9472: ABC214 —— G - Three Permutations
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Description
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 $i$ $(1 \leq i \leq N)$.
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 $i$ $(1 \leq i \leq N)$.
Input
Input is given from Standard Input in the following format:
```
$N$
$p_1$ $\ldots$ $p_N$
$q_1$ $\ldots$ $q_N$
```
```
$N$
$p_1$ $\ldots$ $p_N$
$q_1$ $\ldots$ $q_N$
```
Output
Print the answer.
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.
- $1 \leq p_i, q_i \leq N$
- $p_i \neq p_j \, (i \neq j)$
- $q_i \neq q_j \, (i \neq j)$
- All values in input are integers.
Sample 1 Input
4
1 2 3 4
2 1 4 3
Sample 1 Output
4
There are four valid permutations: (3,4,1,2), (3,4,2,1), (4,3,1,2), and (4,3,2,1).
Sample 2 Input
3
1 2 3
2 1 3
Sample 2 Output
0
The answer may be 0.
Sample 3 Input
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
Sample 3 Output
803776944