Sure, I can help you write a test case generator for the given problem statement. Here is the Python code for generating test cases for the problem:

```python
import random

def generate_permutation(n):
    permutation = list(range(1, n + 1))
    random.shuffle(permutation)
    return permutation

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

# Example of generating a single test case
N, K, P = tcgen()
print(N)
print(K)
print(' '.join(map(str, P)))
```

### Explanation:

1. **generate_permutation(n)**: This helper function creates a random permutation of the set {1, 2, ..., N} by shuffling a list of integers from 1 to N.

2. **tcgen()**: This function generates a random test case:
   - It generates a random integer \(N\) within the range [2, 500,000].
   - It generates a random integer \(K\) within the range [1, N-1].
   - It generates a random permutation of the numbers {1, 2, ..., N} using the `generate_permutation` function.
   
3. **Printing the Test Case**: The function then prints \(N\), \(K\), and the permutation \(P\) in the required format.

You can call the `tcgen()` function multiple times to generate multiple test cases. Each call will provide a new random test case adhering to the constraints specified in the problem statement.

This should give you a comprehensive set of test cases to work with for this competitive programming problem.