[New Algorithm] - Bell Numbers

maths/bell_numbers.py

@@ -0,0 +1,81 @@
+"""
+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
+
+Example:
+To calculate the Bell numbers for sets of lengths from 0 to 5:
+
+>>> import bell_numbers
+>>> bell_numbers.bell_numbers(5)
+[1, 1, 2, 5, 15, 52]
+"""
+
+
+def bell_numbers(n: int) -> list[int]:
+    """
+    Calculate Bell numbers for the sets of lengths from 0 to n. In other words, calculate first (n + 1) Bell numbers.
+
+    Args:
+        n (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 n.
+
+    Examples:
+    >>> bell_numbers(0)
+    [1]
+    >>> bell_numbers(1)
+    [1, 1]
+    >>> bell_numbers(5)
+    [1, 1, 2, 5, 15, 52]
+    """
+    if n < 0:
+        raise ValueError("n must be non-negative")
+
+    bell = [0] * (n + 1)
+    bell[0] = 1
+
+    for i in range(1, n + 1):
+        for j in range(i):
+            bell[i] += _binomial_coefficient(i - 1, j) * bell[j]
+
+    return bell
+
+
+def _binomial_coefficient(n: int, k: int) -> int:
+    """
+    Calculate the binomial coefficient C(n, k) using dynamic programming.
+
+    Args:
+        n (int): Total number of elements.
+        k (int): Number of elements to choose.
+
+    Returns:
+        int: The binomial coefficient C(n, k).
+
+    Examples:
+    >>> _binomial_coefficient(5, 2)
+    10
+    >>> _binomial_coefficient(6, 3)
+    20
+    """
+    if k == 0 or k == n:
+        return 1
+
+    if k > n - k:
+        k = n - k
+
+    coefficient = 1
+    for i in range(k):
+        coefficient *= n - i
+        coefficient //= i + 1
+
+    return coefficient
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()