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

You can define a recursive generator which creates a sub-generator for each 'for loop' which you wish to nest, like this:

def ballsAndBoxes(balls, boxes, boxIndex=0, sumThusFar=0):
    if boxIndex < (boxes - 1):
        for counter in xrange(balls + 1 - sumThusFar):
            for rest in ballsAndBoxes(balls, boxes,
                                      boxIndex + 1,
                                      sumThusFar + counter):
                yield (counter,) + rest
    else:
        yield (balls - sumThusFar,)

When you call this at the top level, it will take only a 'balls' and 'boxes' argument, the others are there as defaults so that the recursive call can pass different things. It will yield tuples of integers (of length 'boxes') that are your values.

To get the exact formatting you specified at the top of this post, you could call it something like this:

BALLS = 8
BOXES = 3
print '\t',
for box in xrange(1, BOXES + 1):
    print '\tbox_%d' % (box,),
print
for position, value in enumerate(ballsAndBoxes(BALLS, BOXES)):
    print 'case-%d\t\t%s' % (position + 1, 
                             "\t".join((str(v) for v in value)))