Score : $600$ points

问题描述

您需要通过以下步骤生成一个图形。

• 选择一个包含 $N$ 个未标记顶点的简单无向图。
• 在图中每个顶点处写入不大于 $K$ 的正整数。这里，不允许存在任何不在任何顶点上书写的不大于 $K$ 的正整数。

求能够得到的图形数量模 $P$ 后的结果。（其中，$P$ 是一个 质数。）

两个图形被认为是相同的，当且仅当可以为每个图形中的顶点标上标签 $v_1, v_2, \dots, v_N$ ，以满足以下条件：

• 对于所有满足 $1 \leq i \leq N$ 的 $i$，两个图形中顶点 $v_i$ 上所写的数字相同。
• 对于所有满足 $1 \leq i < j \leq N$ 的 $i$ 和 $j$，若其中一个图形中 $v_i$ 和 $v_j$ 之间有边，则另一个图形中 $v_i$ 和 $v_j$ 之间也必须有边；反之亦然。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$N$ $K$ $P$

Output

Print the answer.

Sample Input 1

3 1 998244353

Sample Output 1

4

The following four graphs satisfy the condition.

Sample Input 2

3 2 998244353

Sample Output 2

12

The following $12$ graphs satisfy the condition.

Sample Input 3

5 5 998244353

Sample Output 3

1024

Sample Input 4

30 15 202300013

Sample Output 4

62712469

update @ 2024/3/10 11:54:50