Sure! To generate test cases for the problem statement efficiently, we can create a function that randomly generates strings `s` and `t` within the given constraints. Here's a Python function to do that:

```python
import random
import string

def tcgen():
    # Generate lengths for s and t within the given constraints
    len_s = random.randint(1, 3000)
    len_t = random.randint(1, 3000)
    
    # Generate random lowercase strings of the given lengths
    s = ''.join(random.choices(string.ascii_lowercase, k=len_s))
    t = ''.join(random.choices(string.ascii_lowercase, k=len_t))
    
    return s, t

# Example usage:
s, t = tcgen()
print(f"Generated Test Case:\ns = {s}\nt = {t}")
```

This code randomly generates two strings `s` and `t` consisting of lowercase English letters, with lengths between 1 and 3000 inclusive. The function `tcgen` returns these generated strings. You can use this function to generate multiple test cases for the problem.

Feel free to modify the function if you need specific patterns or edge cases. Here's a quick guide on generating some specific edge cases:

1. **Minimal Length Strings**: `len_s` and `len_t` set to 1.
2. **Maximal Length Strings**: `len_s` and `len_t` set to 3000.
3. **Strings With No Common Subsequences**: Generate strings with non-overlapping character sets.
4. **Identical Strings**: Set `s` equal to `t`.

The above function can be adjusted easily to cover these scenarios.