Sure! Let's create a test case generator for the given problem. The generator will produce random values for \(N\), \(K\), and the permutation \(P\) based on the problem constraints.

Here's the code for the test case generator:

```python
import random

def generate_permutation(N):
    P = list(range(1, N + 1))
    random.shuffle(P)
    return P

def tcgen():
    N = random.randint(2, 500000)
    K = random.randint(1, N - 1)
    P = generate_permutation(N)
    
    return N, K, P

def main():
    N, K, P = tcgen()
    print(N, K)
    print(' '.join(map(str, P)))

if __name__ == "__main__":
    main()
```

### Explanation:
1. **generate_permutation(N)**: This function generates a random permutation of the numbers from 1 to \(N\).
2. **tcgen()**:
    - It randomly selects a value for \(N\) within the range [2, 500000].
    - It randomly selects a value for \(K\) within the range [1, \(N-1\)].
    - It generates a random permutation \(P\) of size \(N\).
3. **main()**:
    - Calls the `tcgen()` function to get a random test case.
    - Prints the values of \(N\), \(K\), and the permutation \(P\) in the required format.

You can run this script to generate random test cases for the given problem. Each time you run it, it will produce a new test case. Here are a few example outputs:

### Example Output:
```
4 2
4 3 2 1
```

```
5 1
5 1 3 2 4
```

```
8 3
4 7 8 5 1 6 3 2
```

This way, you can generate an extensive set of test cases to test the robustness and correctness of your solution to the problem.