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Ex - Count Unlabeled Graphs

Score : $600$ points

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.

3 1 998244353

4

The following four graphs satisfy the condition.

3 2 998244353

12

The following $12$ graphs satisfy the condition.

5 5 998244353

1024

30 15 202300013

62712469