Score : $600$ points

问题描述

计算满足以下两个条件的 $N$ 阶非负整数方阵的数量，结果对 $998244353$ 取模：

• 对于所有 $i = 1, 2, \ldots, N$，第 $i$ 行元素之和为 $R_i$；
• 对于所有 $i = 1, 2, \ldots, N$，第 $i$ 列元素之和为 $C_i$。

注意：输入中给出的 $R_i$ 和 $C_i$ 是介于 $0$ 和 $2$ 之间的整数（参见约束条件）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Find the number, modulo $998244353$, of square matrices of size $N$ whose elements are non-negative integers, that satisfy both of the following two conditions:

• for all $i = 1, 2, \ldots, N$, the sum of the elements in the $i$-th row is $R_i$;
• for all $i = 1, 2, \ldots, N$, the sum of the elements in the $i$-th column is $C_i$.

Note that $R_i$ and $C_i$ given in the input are integers between $0$ and $2$ (see Constraints).

Constraints

• $1 \leq N \leq 5000$
• $0 \leq R_i \leq 2$
• $0 \leq C_i \leq 2$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$R_1$ $R_2$ $\ldots$ $R_N$

$C_1$ $C_2$ $\ldots$ $C_N$

Output

Print the answer.

Sample Input 1

3
1 1 1
0 1 2

Sample Output 1

3

The following $3$ matrices satisfy the conditions:

0 1 0
0 0 1
0 0 1

0 0 1
0 1 0
0 0 1

0 0 1
0 0 1
0 1 0

Sample Input 2

3
1 1 1
2 2 2

Sample Output 2

0

Sample Input 3

18
2 0 1 2 0 1 1 2 1 1 2 0 1 2 2 1 0 0
1 1 0 1 1 1 1 1 1 1 1 1 2 1 1 0 2 2

Sample Output 3

968235177

Be sure to print the count modulo $998244353$.

update @ 2024/3/10 11:31:11