To generate test cases for the problem "Partial String Decomposition," we will need to produce valid instances of the input (strings S and T, and integer k) while ensuring they fall within the specified constraints. Here is the test case generator:

```python
import random
import string

def generate_string(length):
    return ''.join(random.choices(string.ascii_lowercase, k=length))

def tcgen():
    # Generate lengths for S and T
    len_S = random.randint(1, 2 * 10**5)
    len_T = random.randint(1, 2 * 10**5)
    
    S = generate_string(len_S)
    T = generate_string(len_T)
    
    # Generate k such that 1 <= k <= len(T)
    k = random.randint(1, len_T)

    return S, T, k

# Example usage
if __name__ == "__main__":
    S, T, k = tcgen()
    print(S)
    print(T)
    print(k)
```

In this generator:
- `generate_string(length)` generates a random string of given length using lowercase alphabets.
- `tcgen()` creates random lengths for strings S and T within the given constraints.
- It generates a random value for `k` ensuring it is between 1 and the length of T.

You can run this script to generate various test cases for the problem.