Problem Description

You are given three positive integers $n$, $m$, and $mod$.

How many ordered $m$-tuples of permutations of $1 \sim n$, $(p_1, p_2, \dots, p_m)$, satisfy:

• Lexicographical order: $p_1 \lt p_2 \lt \dots \lt p_m$.
• Inversion count: $p_1 \gt p_2 \gt \dots \gt p_m$.

Let $f(n,m)$ be the answer modulo $mod$. For all $1 \le i \le n$ and $1 \le j \le m$, output $f(i,j)$.

Input Format

The input contains one line with three positive integers $n$, $m$, $mod$.

Output Format

Output an $n \times m$ matrix, where the entry in row $i$ and column $j$ is $f(i,j)$.

5 3 23333

1 0 0
2 0 0
6 0 0
24 17 0
120 904 1226

Hint

It is guaranteed that $2 \le mod \le 10^9$, $1 \le n \le 15$, and $1 \le m \le 30$. Note that $n$ and $m$ will not simultaneously take the values $15$ and $30$.

The Constraints for $n$ and $m$ are as follows:

• Subtask 1 ($20$ points): $n = 7$, $m = 30$.
• Subtask 2 ($10$ points): $n = 10$, $m = 10$.
• Subtask 3 ($20$ points): $n = 11$, $m = 10$.
• Subtask 4 ($10$ points): $n = 12$, $m = 8$.
• Subtask 5 ($20$ points): $n = 13$, $m = 15$.
• Subtask 6 ($10$ points): $n = 14$, $m = 30$.
• Subtask 7 ($10$ points): $n = 15$, $m = 20$.

Translated by ChatGPT 5