Many Good Tuple Problems

AtCoder

IDabc327_g

Time2000ms

Memory256MB

Difficulty—

> The definition of a good pair of sequences in this problem is the same as in Problem D. A pair of sequences of length $M$ consisting of positive integers at most $N$, $(S, T) = ((S_1, S_2, \dots, S_M), (T_1, T_2, \dots, T_M))$, is said to **be a good pair of sequences** when $(S, T)$ satisfies the following condition. * There exists a sequence $X = (X_1, X_2, \dots, X_N)$ of length $N$ consisting of $0$ and $1$ that satisfies the following condition: * $X_{S_i} \neq X_{T_i}$ for each $i=1, 2, \dots, M$. Among the $N^{2M}$ possible pairs of sequences of length $M$ consisting of positive integers at most $N$, $(A, B) = ((A_1, A_2, \dots, A_M), (B_1, B_2, \dots, B_M))$, find the number, modulo $998244353$, of those that are good pairs of sequences. ## Constraints * $1 \leq N \leq 30$ * $1 \leq M \leq 10^9$ * $N$ and $M$ are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

3 2

Output #1

36

For example, if $A=(1,2), B=(2,3)$, then $(A, B)$ is a good pair of sequences. Indeed, if we set $X=(0,1,0)$, then $X$ is a sequence of length $N$ consisting of $0$ and $1$ that satisfies $X_{A_1} \neq X_{B_1}$ and $X_{A_2} \neq X_{B_2}$. Thus, $(A, B)$ satisfies the condition of being a good pair of sequences.  
There are a total of $36$ good pairs of sequences, so print this number.

Input #2

3 3

Output #2

Input #3

12 34

Output #3

539029838

Input #4

20 231104

Output #4

966200489

API Response (JSON)

{
  "problem": {
    "name": "Many Good Tuple Problems",
    "description": {
      "content": "> The definition of a good pair of sequences in this problem is the same as in Problem D. A pair of sequences of length $M$ consisting of positive integers at most $N$, $(S, T) = ((S_1, S_2, \\dots, 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": "abc327_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "> The definition of a good pair of sequences in this problem is the same as in Problem D.\n\nA pair of sequences of length $M$ consisting of positive integers at most $N$, $(S, T) = ((S_1, S_2, \\dots, S...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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