partition_int_to_n_sums.py

def partition (count, total, steps=1, minimum=0, maximum=None, sum_total=True):
    '''
    @summary:
        Given a number "total" and number of parts to split it to "count", return a count-length tuple made of numbers that sum up to "total".
        Steps defines in what increments to increase each "bin", starting with 0 or minimum sized bins to bins of size total or maximum.
        If sum total is True - only partitions summing to total will return, otherwise, partitions with lower sums will be returned as well.
    @param count:
        How many bins to split total into.
    @param total:
        The total size to split into bins.
    @param steps:
        In what increments to increase the size of the bins.
        i.e. if minimum = 1 and steps = 2, bin sizes will be 1, 3, 5,...
    @param minimum:
        The minimum size of each bin.
    @param maximum:
        The maximum size of each bin. If not passed, it will default to the total.
    @param sum_total:
        If True, only partitions that sum to total will return, otherwise any partitions whose sum is lower than total will be returned.
    @notes:
        Based on https://stackoverflow.com/questions/41837444/how-to-find-all-ways-to-obtain-an-integer-n-as-the-sum-of-m-integers-without-or
    '''
    start = minimum
    maximum = total if (maximum is None or total < maximum) else maximum
    if 0 == count:
        yield tuple([])
    else:
        while (start <= total and
               start <= maximum and
               total <= count * maximum and
               total >= count * minimum):
            for part in partition(count - 1, total - start, steps, minimum, maximum, sum_total):
                res = part + (start,)
                if (not sum_total or len(res) != count or total == sum(res)):
                    yield res
            start = start + steps