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79 lines
2.1 KiB
Python
79 lines
2.1 KiB
Python
"""
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Bell numbers represent the number of ways to partition a set into non-empty
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subsets. This module provides functions to calculate Bell numbers for sets of
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integers. In other words, the first (n + 1) Bell numbers.
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For more information about Bell numbers, refer to:
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https://en.wikipedia.org/wiki/Bell_number
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"""
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def bell_numbers(max_set_length: int) -> list[int]:
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"""
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Calculate Bell numbers for the sets of lengths from 0 to max_set_length.
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In other words, calculate first (max_set_length + 1) Bell numbers.
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Args:
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max_set_length (int): The maximum length of the sets for which
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Bell numbers are calculated.
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Returns:
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list: A list of Bell numbers for sets of lengths from 0 to max_set_length.
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Examples:
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>>> bell_numbers(0)
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[1]
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>>> bell_numbers(1)
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[1, 1]
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>>> bell_numbers(5)
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[1, 1, 2, 5, 15, 52]
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"""
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if max_set_length < 0:
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raise ValueError("max_set_length must be non-negative")
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bell = [0] * (max_set_length + 1)
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bell[0] = 1
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for i in range(1, max_set_length + 1):
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for j in range(i):
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bell[i] += _binomial_coefficient(i - 1, j) * bell[j]
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return bell
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def _binomial_coefficient(total_elements: int, elements_to_choose: int) -> int:
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"""
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Calculate the binomial coefficient C(total_elements, elements_to_choose)
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Args:
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total_elements (int): The total number of elements.
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elements_to_choose (int): The number of elements to choose.
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Returns:
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int: The binomial coefficient C(total_elements, elements_to_choose).
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Examples:
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>>> _binomial_coefficient(5, 2)
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10
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>>> _binomial_coefficient(6, 3)
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20
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"""
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if elements_to_choose in {0, total_elements}:
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return 1
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if elements_to_choose > total_elements - elements_to_choose:
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elements_to_choose = total_elements - elements_to_choose
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coefficient = 1
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for i in range(elements_to_choose):
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coefficient *= total_elements - i
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coefficient //= i + 1
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return coefficient
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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