Row Column Sums 2

AtCoder

IDabc273_g

Time3000ms

Memory256MB

Difficulty—

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: $N$ $R_1$ $R_2$ $\ldots$ $R_N$ $C_1$ $C_2$ $\ldots$ $C_N$ [samples]

Samples

Input #1

3
1 1 1
0 1 2

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

Input #2

3
1 1 1
2 2 2

Output #2

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

Output #3

968235177

Be sure to print the count modulo $998244353$.

API Response (JSON)

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    "name": "Row Column Sums 2",
    "description": {
      "content": "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",
      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc273_g"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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:\n\n*   for all $i = 1, 2, \\ldots, N$, the...",
      "is_translate": false,
      "language": "English"
    }
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