Large Heap

AtCoder

IDagc060_c

Time10000ms

Memory256MB

Difficulty—

Consider a permutation $P=(P_1,P_2,\cdots,P_{2^N-1})$ of $(1,2,\cdots,2^N-1)$. $P$ is said to be **heaplike** if and only if all of the following conditions are satisfied. * $P_i < P_{2i}$ ($1 \leq i \leq 2^{N-1}-1$) * $P_i < P_{2i+1}$ ($1 \leq i \leq 2^{N-1}-1$) You are given integers $A$ and $B$. Let $U=2^A$ and $V=2^{B+1}-1$. Find the probability, modulo $998244353$, that $P_U<P_V$ for a heaplike permutation chosen uniformly at random. Definition of probability modulo $998244353$ It can be proved that the sought probability is always rational. Additionally, under the constraints of this problem, when the sought rational number is represented as an irreducible fraction $\frac{P}{Q}$, it can be proved that $Q \neq 0 \pmod{998244353}$. Therefore, there is a unique integer $R$ such that $R \times Q \equiv P \pmod{998244353}$ and $0 \leq R \lt 998244353$. Print this $R$. ## Constraints * $2 \leq N \leq 5000$ * $1 \leq A,B \leq N-1$ * All numbers in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $A$ $B$ [samples]

Samples

Input #1

2 1 1

Output #1

499122177

There are two heaplike permutations: $P=(1,2,3),(1,3,2)$. The probability that $P_2<P_3$ is $1/2$.

Input #2

3 1 2

Output #2

124780545

Input #3

4 3 2

Output #3

260479386

Input #4

2022 12 25

Output #4

741532295

API Response (JSON)

{
  "problem": {
    "name": "Large Heap",
    "description": {
      "content": "Consider a permutation $P=(P_1,P_2,\\cdots,P_{2^N-1})$ of $(1,2,\\cdots,2^N-1)$. $P$ is said to be **heaplike** if and only if all of the following conditions are satisfied. *   $P_i < P_{2i}$ ($1 \\leq",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 10000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc060_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Consider a permutation $P=(P_1,P_2,\\cdots,P_{2^N-1})$ of $(1,2,\\cdots,2^N-1)$. $P$ is said to be **heaplike** if and only if all of the following conditions are satisfied.\n\n*   $P_i < P_{2i}$ ($1 \\leq...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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