Python/maths/special_numbers/bell_numbers.py

"""
Bell numbers represent the number of ways to partition a set into non-empty
subsets. This module provides functions to calculate Bell numbers for sets of
integers. In other words, the first (n + 1) Bell numbers.

For more information about Bell numbers, refer to:
https://en.wikipedia.org/wiki/Bell_number
"""


def bell_numbers(max_set_length: int) -> list[int]:
    """
    Calculate Bell numbers for the sets of lengths from 0 to max_set_length.
    In other words, calculate first (max_set_length + 1) Bell numbers.

    Args:
        max_set_length (int): The maximum length of the sets for which
        Bell numbers are calculated.

    Returns:
        list: A list of Bell numbers for sets of lengths from 0 to max_set_length.

    Examples:
    >>> bell_numbers(0)
    [1]
    >>> bell_numbers(1)
    [1, 1]
    >>> bell_numbers(5)
    [1, 1, 2, 5, 15, 52]
    """
    if max_set_length < 0:
        raise ValueError("max_set_length must be non-negative")

    bell = [0] * (max_set_length + 1)
    bell[0] = 1

    for i in range(1, max_set_length + 1):
        for j in range(i):
            bell[i] += _binomial_coefficient(i - 1, j) * bell[j]

    return bell


def _binomial_coefficient(total_elements: int, elements_to_choose: int) -> int:
    """
    Calculate the binomial coefficient C(total_elements, elements_to_choose)

    Args:
        total_elements (int): The total number of elements.
        elements_to_choose (int): The number of elements to choose.

    Returns:
        int: The binomial coefficient C(total_elements, elements_to_choose).

    Examples:
    >>> _binomial_coefficient(5, 2)
    10
    >>> _binomial_coefficient(6, 3)
    20
    """
    if elements_to_choose in {0, total_elements}:
        return 1

    if elements_to_choose > total_elements - elements_to_choose:
        elements_to_choose = total_elements - elements_to_choose

    coefficient = 1
    for i in range(elements_to_choose):
        coefficient *= total_elements - i
        coefficient //= i + 1

    return coefficient


if __name__ == "__main__":
    import doctest

    doctest.testmod()