Enumeration of combinations of N balls in A boxes?

Question

I want to enumerate all possible combinations of N balls in A boxes.

example: I have 8

Answer 1

See itertools.combinations_with_replacement in 3.1 for an example written in python. Additionally, it's common in combinatorics to transform a combination-with-replacement problem into the usual combination-without-replacement problem, which is already builtin in 2.6 itertools. This has the advantage of not generating discarded tuples, like solutions based on product or permutation. Here's an example using the standard (n, r) terminology, which would be (A, N) in your example.

import itertools, operator
def combinations_with_replacement_counts(n, r):
    size = n + r - 1
    for indices in itertools.combinations(range(size), n-1):
        starts = [0] + [index+1 for index in indices]
        stops = indices + (size,)
        yield tuple(map(operator.sub, stops, starts))

>>> list(combinations_with_replacement_counts(3, 8))
[(0, 0, 8), (0, 1, 7), (0, 2, 6), (0, 3, 5), (0, 4, 4), (0, 5, 3), (0, 6, 2), (0, 7, 1), (0, 8, 0), (1, 0, 7), (1, 1, 6), (1, 2, 5), (1, 3, 4), (1, 4, 3), (1, 5, 2), (1, 6, 1), (1, 7, 0), (2, 0, 6), (2, 1, 5), (2, 2, 4), (2, 3, 3), (2, 4, 2), (2, 5, 1), (2, 6, 0), (3, 0, 5), (3, 1, 4), (3, 2, 3), (3, 3, 2), (3, 4, 1), (3, 5, 0), (4, 0, 4), (4, 1, 3), (4, 2, 2), (4, 3, 1), (4, 4, 0), (5, 0, 3), (5, 1, 2), (5, 2, 1), (5, 3, 0), (6, 0, 2), (6, 1, 1), (6, 2, 0), (7, 0, 1), (7, 1, 0), (8, 0, 0)]