[New Algorithm] - Bell Numbers (#9324)

* Add Bell Numbers

* Use descriptive variable names

* Add type hints

* Fix failing tests

maths/bell_numbers.py

@@ -0,0 +1,78 @@
+"""
+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()