Ex - Count Unlabeled Graphs

AtCoder

IDabc284_h

Time2000ms

Memory256MB

Difficulty—

You are to generate a graph by the following procedure. * Choose a simple undirected graph with $N$ unlabeled vertices. * Write a positive integer at most $K$ in each vertex in the graph. Here, there must not be a positive integer at most $K$ that is not written in any vertex. Find the number of possible graphs that can be obtained, modulo $P$. ($P$ is a **prime**.) Two graphs are considered the same if and only if one can label the vertices in each graph as $v_1, v_2, \dots, v_N$ to satisfy the following conditions. * For every $i$ such that $1 \leq i \leq N$, the numbers written in vertex $v_i$ in the two graphs are the same. * For every $i$ and $j$ such that $1 \leq i \lt j \leq N$, there is an edge between $v_i$ and $v_j$ in one of the graphs if and only if there is an edge between $v_i$ and $v_j$ in the other graph. ## Constraints * $1 \leq K \leq N \leq 30$ * $10^8 \leq P \leq 10^9$ * $P$ is a prime. * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ $P$ [samples]

Samples

Input #1

3 1 998244353

Output #1

4

The following four graphs satisfy the condition.
![image](https://img.atcoder.jp/ghi/abc283h_43c4abe0e541b7ebeaa8db2854cece91caeca71f03f452ca13c11e82f85e3a56.png)

Input #2

3 2 998244353

Output #2

12

The following $12$ graphs satisfy the condition.
![image](https://img.atcoder.jp/ghi/abc284h2_ca96b7cb451b0e495209e3e201576d278de3fb823e5d2404bbce5d9f704e3259.png)

Input #3

5 5 998244353

Output #3

Input #4

30 15 202300013

Output #4

62712469

API Response (JSON)

{
  "problem": {
    "name": "Ex - Count Unlabeled Graphs",
    "description": {
      "content": "You are to generate a graph by the following procedure. *   Choose a simple undirected graph with $N$ unlabeled vertices. *   Write a positive integer at most $K$ in each vertex in the graph. Here, t",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc284_h"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are to generate a graph by the following procedure.\n\n*   Choose a simple undirected graph with $N$ unlabeled vertices.\n*   Write a positive integer at most $K$ in each vertex in the graph. Here, t...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments