Select and Split

AtCoder

IDarc173_f

Time2000ms

Memory256MB

Difficulty—

There is a blackboard with a set of positive integers written on it. Initially, the set $S=\lbrace 1,2,\dots,A,A+1,A+2,\dots,A+B\rbrace$ is written on the blackboard. Takahashi wants to perform the following operation $N-1$ times to have $N$ sets on the blackboard: * From the sets of integers written on the blackboard, choose a set $S_0$ that contains at least one element not greater than $A$ and at least one element not less than $A+1$. From the chosen set $S_0$, choose one element $a$ not greater than $A$ and one element $b$ not less than $A+1$. Erase the set $S_0$ from the blackboard and write two sets $S_1$ and $S_2$ of his choice that satisfy all of the following conditions: * The union of $S_1$ and $S_2$ is $S_0$, and they have no common elements. * $a \in S_1, b \in S_2$ Find the number of possible ways to perform the series of operations, modulo $998244353$. Here, two series of operations are distinguished when there is an $i\ (1 \leq i \leq N-1)$ such that $S_0$, $a$, $b$, $S_1$, or $S_2$ chosen in the $i$\-th operation of one series is different from that of the other series. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq A,B \leq 2 \times 10^5$ * $N \leq A+B$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A$ $B$ [samples]

Samples

Input #1

3 2 4

Output #1

1728

One possible series of operations is as follows:

*   Choose $S_0=\lbrace 1,2,3,4,5,6\rbrace$, let $a=2$ and $b=5$, and let $S_1 =\lbrace 1,2,3,6\rbrace$ and $S_2=\lbrace 4,5\rbrace$. There are now two sets of integers written on the blackboard: $\lbrace 1,2,3,6\rbrace$ and $\lbrace 4,5\rbrace$.
*   Choose $S_0=\lbrace 1,2,3,6\rbrace$, let $a=1$ and $b=3$, and let $S_1 = \lbrace1,2\rbrace$ and $S_2=\lbrace 3,6\rbrace$. There are now three sets of integers written on the blackboard: $\lbrace 1,2\rbrace$, $\lbrace 3,6\rbrace$, and $\lbrace 4,5\rbrace$.

Input #2

4 1 3

Output #2

6

If we let $a=1$ and $b=2$ and let $S_1 = \lbrace 1\rbrace$ and $S_2 = \lbrace 2,3,4\rbrace$ in the first operation, we can no longer perform the second and subsequent operations.
Do not count these cases where we cannot complete $N-1$ operations.

Input #3

5 6 6

Output #3

84486693

Input #4

173173 173173 173173

Output #4

446948086

API Response (JSON)

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  "problem": {
    "name": "Select and Split",
    "description": {
      "content": "There is a blackboard with a set of positive integers written on it. Initially, the set $S=\\lbrace 1,2,\\dots,A,A+1,A+2,\\dots,A+B\\rbrace$ is written on the blackboard. Takahashi wants to perform the fo",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc173_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a blackboard with a set of positive integers written on it. Initially, the set $S=\\lbrace 1,2,\\dots,A,A+1,A+2,\\dots,A+B\\rbrace$ is written on the blackboard.\nTakahashi wants to perform the fo...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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