Brute force calculator for https://math.stackexchange.com/questions/3039802/coloured-partitions/3042074

ColouredPartitions.py

from typing import List


def generate_set(half_size: int) -> List[str]:
    oranges: List[str] = [f"B{i + 1}" for i in range(half_size)]
    blues: List[str] = [f"O{i + 1}" for i in range(half_size)]
    return oranges + blues


def partition(collection):
    # "borrowed" from https://stackoverflow.com/questions/19368375/set-partitions-in-python
    if len(collection) == 1:
        yield [collection]
        return

    first = collection[0]
    for smaller in partition(collection[1:]):
        # insert `first` in each of the subpartition's subsets
        for n, subset in enumerate(smaller):
            yield smaller[:n] + [[first] + subset] + smaller[n + 1:]
        # put `first` in its own subset
        yield [[first]] + smaller


def is_partition_good(partition: List[List[str]]) -> bool:
    def is_subset_good(subset: List[str]) -> bool:
        if len(subset) < 2:
            return False
        first_letters: List[str] = [word[0] for word in subset]
        return any(map(lambda x: x == 'B', first_letters))

    for subset in partition:
        if not is_subset_good(subset):
            return False
    return True


if __name__ == "__main__":
    for x in range(1, 6):
        print(f"{x}:\t{len(list(filter(is_partition_good, list(partition(generate_set(x))))))}")