Python/dynamic_programming/integer_partition.py

"""
The number of partitions of a number n into at least k parts equals the number of
partitions into exactly k parts plus the number of partitions into at least k-1 parts.
Subtracting 1 from each part of a partition of n into k parts gives a partition of n-k
into k parts. These two facts together are used for this algorithm.
* https://en.wikipedia.org/wiki/Partition_(number_theory)
* https://en.wikipedia.org/wiki/Partition_function_(number_theory)
"""


def partition(m: int) -> int:
    """
    >>> partition(5)
    7
    >>> partition(7)
    15
    >>> partition(100)
    190569292
    >>> partition(1_000)
    24061467864032622473692149727991
    >>> partition(-7)
    Traceback (most recent call last):
        ...
    IndexError: list index out of range
    >>> partition(0)
    Traceback (most recent call last):
        ...
    IndexError: list assignment index out of range
    >>> partition(7.8)
    Traceback (most recent call last):
        ...
    TypeError: 'float' object cannot be interpreted as an integer
    """
    memo: list[list[int]] = [[0 for _ in range(m)] for _ in range(m + 1)]
    for i in range(m + 1):
        memo[i][0] = 1

    for n in range(m + 1):
        for k in range(1, m):
            memo[n][k] += memo[n][k - 1]
            if n - k > 0:
                memo[n][k] += memo[n - k - 1][k]

    return memo[m][m - 1]


if __name__ == "__main__":
    import sys

    if len(sys.argv) == 1:
        try:
            n = int(input("Enter a number: ").strip())
            print(partition(n))
        except ValueError:
            print("Please enter a number.")
    else:
        try:
            n = int(sys.argv[1])
            print(partition(n))
        except ValueError:
            print("Please pass a number.")