maths.special_numbers.bell_numbers
==================================

.. py:module:: maths.special_numbers.bell_numbers

.. autoapi-nested-parse::

   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



Functions
---------

.. autoapisummary::

   maths.special_numbers.bell_numbers._binomial_coefficient
   maths.special_numbers.bell_numbers.bell_numbers


Module Contents
---------------

.. py:function:: _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


.. py:function:: 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(-2)
   Traceback (most recent call last):
       ...
   ValueError: max_set_length must be non-negative
   >>> bell_numbers(0)
   [1]
   >>> bell_numbers(1)
   [1, 1]
   >>> bell_numbers(5)
   [1, 1, 2, 5, 15, 52]