Double Landscape

AtCoder

IDkeyence2019_d

Time2000ms

Memory256MB

Difficulty—

Consider writing each of the integers from $1$ to $N \times M$ in a grid with $N$ rows and $M$ columns, without duplicates. Takahashi thinks it is not fun enough, and he will write the numbers under the following conditions: * The largest among the values in the $i$\-th row $(1 \leq i \leq N)$ is $A_i$. * The largest among the values in the $j$\-th column $(1 \leq j \leq M)$ is $B_j$. For him, find the number of ways to write the numbers under these conditions, modulo $10^9 + 7$. ## Constraints * $1 \leq N \leq 1000$ * $1 \leq M \leq 1000$ * $1 \leq A_i \leq N \times M$ * $1 \leq B_j \leq N \times M$ * $A_i$ and $B_j$ are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $A_1$ $A_2$ $...$ $A_{N}$ $B_1$ $B_2$ $...$ $B_{M}$ [samples]

Samples

Input #1

2 2
4 3
3 4

Output #1

2

$(A_1, A_2) = (4, 3)$ and $(B_1, B_2) = (3, 4)$. In this case, there are two ways to write the numbers, as follows:

*   $1$ in $(1, 1)$, $4$ in $(1, 2)$, $3$ in $(2, 1)$ and $2$ in $(2, 2)$.
*   $2$ in $(1, 1)$, $4$ in $(1, 2)$, $3$ in $(2, 1)$ and $1$ in $(2, 2)$.

Here, $(i, j)$ denotes the square at the $i$\-th row and the $j$\-th column.

Input #2

3 3
5 9 7
3 6 9

Output #2

0

Since there is no way to write the numbers under the condition, $0$ should be printed.

Input #3

2 2
4 4
4 4

Output #3

Input #4

14 13
158 167 181 147 178 151 179 182 176 169 180 129 175 168
181 150 178 179 167 180 176 169 182 177 175 159 173

Output #4

343772227

API Response (JSON)

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  "problem": {
    "name": "Double Landscape",
    "description": {
      "content": "Consider writing each of the integers from $1$ to $N \\times M$ in a grid with $N$ rows and $M$ columns, without duplicates. Takahashi thinks it is not fun enough, and he will write the numbers under t",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "keyence2019_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Consider writing each of the integers from $1$ to $N \\times M$ in a grid with $N$ rows and $M$ columns, without duplicates. Takahashi thinks it is not fun enough, and he will write the numbers under t...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments