Weird LIS

AtCoder

IDagc055_c

Time2000ms

Memory256MB

Difficulty—

You are given integers $N$ and $M$. Find the number of arrays $A=[A_1, A_2, \ldots, A_N]$ of length $N$ such that the following conditions hold: * $2 \le A_i \le M$ ($1 \leq i \leq N$) * There exists a permutation $P=[P_1,P_2,\ldots,P_N]$ of integers from $1$ to $N$ with the following property: * For every $i$ from $1$ to $N$, $A_i$ equals the length of the longest increasing subsequence of the sequence $[P_1, P_2, \ldots, P_{i-1}, P_{i+1}, \ldots, P_{N-1}, P_N]$. As this number can be very large, output it modulo some prime $Q$. ## Constraints * $3 \le N \le 5000$. * $2 \le M \le N-1$. * $10^8 \le Q \le 10^9$ * $Q$ is a prime. ## Input Input is given from Standard Input in the following format: $N$ $M$ $Q$ [samples]

Samples

Input #1

3 2 686926217

Output #1

1

The only such array is $[2, 2, 2]$, for which exists a permutation $[1, 2, 3]$.

Input #2

4 3 354817471

Output #2

9

There are $9$ such arrays: $[2, 2, 2, 2]$, $[2, 2, 2, 3]$, $[2, 2, 3, 2]$, $[2, 2, 3, 3]$, $[2, 3, 2, 2]$, $[2, 3, 3, 2]$, $[3, 2, 2, 2]$, $[3, 3, 2, 2]$, $[3, 3, 3, 3]$.

Input #3

5 2 829412599

Output #3

1

The only such array is $[2, 2, 2, 2, 2]$.

Input #4

5 3 975576997

Output #4

Input #5

69 42 925171057

Output #5

801835311

API Response (JSON)

{
  "problem": {
    "name": "Weird LIS",
    "description": {
      "content": "You are given integers $N$ and $M$. Find the number of arrays $A=[A_1, A_2, \\ldots, A_N]$ of length $N$ such that the following conditions hold: *   $2 \\le A_i \\le M$ ($1 \\leq i \\leq N$) *   There ex",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc055_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given integers $N$ and $M$. Find the number of arrays $A=[A_1, A_2, \\ldots, A_N]$ of length $N$ such that the following conditions hold:\n\n*   $2 \\le A_i \\le M$ ($1 \\leq i \\leq N$)\n*   There ex...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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