Sure! To write a test case generator for the given problem, we will need to generate:

1. An integer \( N \) (2 ≤ \( N \) ≤ 500,000), representing the size of the permutation.
2. An integer \( K \) (1 ≤ \( K \) ≤ \( N-1 \)).
3. A permutation \( P \) of the set {1, 2, ..., \( N \)}.

Here's a Python function to generate a random test case for this problem:

```python
import random

def generate_test_case():
    # Generate random N between 2 and 500,000
    N = random.randint(2, 500000)
    
    # Generate random K between 1 and N-1
    K = random.randint(1, N-1)
    
    # Generate a permutation of the set {1, 2, ..., N}
    P = list(range(1, N + 1))
    random.shuffle(P)
    
    return N, K, P

# Example usage:
N, K, P = generate_test_case()
print(N)
print(K)
print(' '.join(map(str, P)))
```

This function `generate_test_case` will:

1. Randomly generate an integer \( N \) between 2 and 500,000.
2. Randomly generate an integer \( K \) between 1 and \( N-1 \).
3. Create a permutation of the set \{1, 2, ..., \( N \)\}.
4. Shuffle the permutation to ensure it is random.

To use the test case generator, you can call `generate_test_case()` which will return the values of \( N \), \( K \), and the permutation \( P \). The example usage provided will print the generated values in the expected format.