Nondivisible Prefix Sums

AtCoder

IDagc052_c

Time2000ms

Memory256MB

Difficulty—

You are given a prime number $P$, which you don't like. Let's call an array of integers $A_1, A_2, \dots, A_N$ **good**, if it is possible to reorder the elements in such a way that no prefix sum is divisible by $P$ (that is, there is **no** $i$ with $1 \le i \le N$ and $A_1 + A_2 + \dots + A_i \equiv 0 \bmod P$ after the reordering). Consider all $(P-1)^N$ arrays of length $N$ with elements from $1$ to $P-1$. How many of them are **good**? As this number can be very big, output it modulo $998244353$. ## Constraints * $1 \le N \le 5000$ * $2 \le P \le 10^8$ * $P$ is a prime. ## Input Input is given from Standard Input in the following format: $N$ $P$ [samples]

Samples

Input #1

2 5

Output #1

12

There are $12$ good arrays: $[1, 1]$, $[1, 2]$, $[1, 3]$, $[2, 1]$, $[2, 2]$, $[2, 4]$, $[3, 1]$, $[3, 3]$, $[3, 4]$, $[4, 2]$, $[4, 3]$, $[4, 4]$.

Input #2

4 3

Output #2

8

There are $8$ good arrays: $[1, 1, 1, 2]$, $[1, 1, 2, 1]$, $[1, 2, 1, 1]$, $[2, 1, 1, 1]$, $[2, 2, 2, 1$\], $[2, 2, 1, 2]$, $[2, 1, 2, 2]$, $[1, 2, 2, 2]$.

Input #3

5000 99999989

Output #3

51699346

Input #4

2021 307

Output #4

644635349

API Response (JSON)

{
  "problem": {
    "name": "Nondivisible Prefix Sums",
    "description": {
      "content": "You are given a prime number $P$, which you don't like. Let's call an array of integers $A_1, A_2, \\dots, A_N$ **good**, if it is possible to reorder the elements in such a way that no prefix sum is d",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc052_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a prime number $P$, which you don't like.\nLet's call an array of integers $A_1, A_2, \\dots, A_N$ **good**, if it is possible to reorder the elements in such a way that no prefix sum is d...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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