Flipping Coins

AtCoder

IDarc134_f

Time5000ms

Memory256MB

Difficulty—

There are $N$ coins numbered $1, 2, \ldots, N$ arranged in a row. Initially, all coins face up. Snuke chooses a permutation $p$ of $(1,\ldots,N)$ with equal probability, and do $N$ operations. The $i$\-th operation does the following. * If the coin numbered $i$ faces down, do nothing. * If the coin numbered $i$ faces up, turn that coin over and then turn over the coin numbered $p_i$. After the $N$ operations, Snuke will receive $W^k$ yen (Japanese currency) from his mother, where $k$ is the number of coins facing up. Find the expected value of the amount Snuke will get, multiplied by $N!$ (it can be proved that this is an integer), modulo $998244353$. ## Constraints * All values in input are integers. * $1 \leq N \leq 2 \times 10^5$ * $1 \leq W < 998244353$ ## Input Input is given from Standard Input in the following format: $N$ $W$ [samples]

Samples

Input #1

4 2

Output #1

90

*   When $p=(1,4,2,3)$, the operations go as follows.
    *   In the $1$\-st operation, the coin numbered $1$ faces down, and then the coin numbered $1$ faces up.
    *   In the $2$\-nd operation, the coin numbered $2$ faces down, and then the coin numbered $4$ faces down.
    *   In the $3$\-rd operation, the coin numbered $3$ faces down, and then the coin numbered $2$ faces up.
    *   In the $4$\-th operation, nothing is done.
*   In the end, two coins face up, so he receives $4$ yen.

Input #2

2 100

Output #2

Input #3

200000 12345

Output #3

541410753

*   Be sure to find the value modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Flipping Coins",
    "description": {
      "content": "There are $N$ coins numbered $1, 2, \\ldots, N$ arranged in a row. Initially, all coins face up. Snuke chooses a permutation $p$ of $(1,\\ldots,N)$ with equal probability, and do $N$ operations. The $i$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc134_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ coins numbered $1, 2, \\ldots, N$ arranged in a row. Initially, all coins face up.\nSnuke chooses a permutation $p$ of $(1,\\ldots,N)$ with equal probability, and do $N$ operations. The $i$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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