Sets Scores

AtCoder

IDarc147_d

Time2000ms

Memory256MB

Difficulty—

Consider a sequence of integer sets of length $N$: $S=(S_1,S_2,\dots,S_N)$. We call a sequence **brilliant** if it satisfies all of the following conditions: * $S_i$ is a (possibly empty) integer set, and its elements are in the range $[1,M]$. $(1 \le i \le N)$ * The number of integers that is included in exactly one of $S_i$ and $S_{i+1}$ is $1$. $(1 \le i \le N-1)$ We define the score of a brilliant sequence $S$ as $\displaystyle \prod_{i=1}^{M}$ $($ the number of sets among $S_1,S_2,\dots,S_N$ that include $i$.$)$. Find, modulo $998244353$, the sum of the scores of all possible brilliant sequences. ## Constraints * $1 \le N,M \le 2 \times 10^5$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 3

Output #1

24

Among all possible brilliant sequences, the following $6$ have positive scores.

*   $S_1={1,2},S_2={1,2,3}$
*   $S_1={1,3},S_2={1,2,3}$
*   $S_1={2,3},S_2={1,2,3}$
*   $S_1={1,2,3},S_2={1,2}$
*   $S_1={1,2,3},S_2={1,3}$
*   $S_1={1,2,3},S_2={2,3}$

All of them have a score of $4$, so the sum of them is $24$.

Input #2

12 34

Output #2

786334067

API Response (JSON)

{
  "problem": {
    "name": "Sets Scores",
    "description": {
      "content": "Consider a sequence of integer sets of length $N$: $S=(S_1,S_2,\\dots,S_N)$. We call a sequence **brilliant** if it satisfies all of the following conditions: *   $S_i$ is a (possibly empty) integer s",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc147_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Consider a sequence of integer sets of length $N$: $S=(S_1,S_2,\\dots,S_N)$. We call a sequence **brilliant** if it satisfies all of the following conditions:\n\n*   $S_i$ is a (possibly empty) integer s...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments