Sure, I can write a test case generator for the given problem statement. Below is the Python code to generate random test cases for the problem "Takahashi's Tap Dance":

```python
import random

def tcgen():
    # Length of the string S, between 1 and 100 inclusive
    length = random.randint(1, 100)
    
    # Characters in S are L, R, U, or D
    characters = ['L', 'R', 'U', 'D']
    
    # Generate a random string of the specified length using the characters
    S = ''.join(random.choice(characters) for _ in range(length))
    
    return S

# Example usage
if __name__ == "__main__":
    for _ in range(5):  # Generate 5 test cases
        print(tcgen())
```

### Explanation:
- The `tcgen` function generates a random test case for the problem.
- The length of the string \( S \) is randomly chosen between 1 and 100 inclusive.
- Each character in the string \( S \) is randomly chosen from the set {'L', 'R', 'U', 'D'}.
- The function returns the generated string \( S \).

### Sample Output:
Running the `tcgen` function multiple times might produce outputs like:
```
LURD
ULRUD
DDDUUU
LRLRLR
URDL
```

These outputs represent different possible dance instructions for Takahashi according to the constraints and format described in the problem statement.