Not So Consecutive

AtCoder

IDarc169_c

Time2000ms

Memory256MB

Difficulty—

You are given an integer $N$. An integer sequence $x=(x_1,x_2,\cdots,x_N)$ of length $N$ is called a **good** sequence if and only if the following conditions are satisfied: * Each element of $x$ is an integer between $1$ and $N$, inclusive. * For each integer $i$ ($1 \leq i \leq N$), there is no position in $x$ where $i$ appears $i+1$ or more times in a row. You are given an integer sequence $A=(A_1,A_2,\cdots,A_N)$ of length $N$. Each element of $A$ is $-1$ or an integer between $1$ and $N$. Find the number, modulo $998244353$, of good sequences that can be obtained by replacing each $-1$ in $A$ with an integer between $1$ and $N$. ## Constraints * $1 \leq N \leq 5000$ * $A_i=-1$ or $1 \leq A_i \leq N$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples]

Samples

Input #1

2
-1 -1

Output #1

3

You can obtain four sequences by replacing each $-1$ with an integer between $1$ and $2$.
$A=(1,1)$ is not a good sequence because $1$ appears twice in a row.
The other sequences $A=(1,2),(2,1),(2,2)$ are good.
Thus, the answer is $3$.

Input #2

3
2 -1 2

Output #2

Input #3

4
-1 1 1 -1

Output #3

Input #4

20
9 -1 -1 -1 -1 -1 -1 -1 -1 -1 7 -1 -1 -1 19 4 -1 -1 -1 -1

Output #4

128282166

API Response (JSON)

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    "name": "Not So Consecutive",
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      "content": "You are given an integer $N$. An integer sequence $x=(x_1,x_2,\\cdots,x_N)$ of length $N$ is called a **good** sequence if and only if the following conditions are satisfied: *   Each element of $x$ i",
      "description_type": "Markdown"
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    "difficulty": "None",
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    "sign": "arc169_c"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given an integer $N$. An integer sequence $x=(x_1,x_2,\\cdots,x_N)$ of length $N$ is called a **good** sequence if and only if the following conditions are satisfied:\n\n*   Each element of $x$ i...",
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