Description

Let $p$ be a prime number.

You are given an undirected complete graph with $p^k$ vertices (there is an undirected edge between any two vertices). The vertices are labeled from $0$ to $p^k-1$.

Now you need to find some complete subgraphs on $p$ vertices such that every edge in the original graph belongs to one and only one of these complete subgraphs.

Obviously, the number of complete subgraphs you need to find is $\frac{p^k(p^k-1)/2}{p(p-1)/2}$. You can see that this expression is always an integer.

Input Format

One line containing two positive integers $p, k$.

Output Format

If there is no solution, output one line NO.

Otherwise, output one line YES, then output $\frac{p^k(p^k-1)/2}{p(p-1)/2}$ lines. Each line contains $p$ numbers, representing the vertex labels of one complete subgraph you found.

You may output any valid solution in any order.

2 2

YES
0 1
2 0
3 0
1 2
1 3
3 2

3 1

YES
0 1 2

Hint

For $10\%$ of the testdata, $k \le 1$.

For $50\%$ of the testdata, $k \le 2$.

Another $20\%$ of the testdata has $p = 2$.

For $100\%$ of the testdata, $k$ is a positive integer, $p$ is prime, and $2 \le p^k \le 2000$.

In addition, the total output size is guaranteed not to exceed 2MB, but you should still pay attention to the time spent on output.

Translated by ChatGPT 5