Montage

AtCoder

IDarc162_f

Time3000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $M$. Among the $2^{NM}$ matrices $A$ with $N$ rows and $M$ columns where each element is $0$ or $1$, find the number, modulo $998244353$, of ones that satisfy the following condition: * $A_{a, b} \times A_{c, d} \leq A_{a, d} \times A_{c, b}$ for every quadruple of integers $(a, b, c, d)$ such that $1 \leq a < c \leq N$ and $1 \leq b < d \leq M$. ## Constraints * $1 \leq N, M \leq 400$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 2

Output #1

13

The condition is $A_{1,1} \times A_{2,2} \leq A_{1,2} \times A_{2,1}$. All $13$ matrices other than $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ satisfy the condition.

Input #2

1 30

Output #2

75497471

All $2^{NM}$ matrices satisfy the condition, so print $2^{30}$ modulo $998244353$, that is, $75497471$.

Input #3

400 400

Output #3

412670892

API Response (JSON)

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  "problem": {
    "name": "Montage",
    "description": {
      "content": "You are given positive integers $N$ and $M$. Among the $2^{NM}$ matrices $A$ with $N$ rows and $M$ columns where each element is $0$ or $1$, find the number, modulo $998244353$, of ones that satisfy t",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc162_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $M$. Among the $2^{NM}$ matrices $A$ with $N$ rows and $M$ columns where each element is $0$ or $1$, find the number, modulo $998244353$, of ones that satisfy t...",
      "is_translate": false,
      "language": "English"
    }
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}

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