All possible ways to place colored balls into bins

Question

I have a number of bins. Each bin accommodates only one ball.

Let's say, I have

Na number of red balls

Nb number of blue balls

Nc number of green balls

and so on.

I want to find out all possible ways of placing the balls into bins of length (Na+Nb+Nc).

For example, let's say I have only two red balls and two blue balls. I want to place them into 4 bins. The possible ways of arrangements are:

R R B B
R B R B
R B B R
B R R B
B R B R
B B R R

( I hope I did not miss any combination)

Is there any easy way to generate the indices for the different colors, for example:

First row is : R=(0,1) B=(2,3)
Second row is : R=(0,2) B=(1,3)

Is there an easy way to generate this in numpy?

The bins actually have different weights, something like:

[0.1, 0.3, 0.2, 0.5]

So for the combination R R B B represented as R at (0,1) and B at (2,3) , the total weight for R is 0.1+0.3=0.4 and for B is 0.2+0.5=0.7

I ultimately am interested in the total weight for each color in the different arrangements and want to choose the best one from another cost function f(total_weight(R), total_weight(B)). Any comments if the generation of total weight could be done in a different easy way in numpy?

Answer 1

Here is one possible way of implementing:

def permutations(colors, l):
    if l == 0:
        yield []
    else:
        for i in range(len(colors)):
            if colors[i]:
                la = colors[:i]
                lb = [colors[i][1:]]
                lc = colors[i + 1:]
                choice = colors[i][0]
                for rest in permutations(la + lb + lc, l - 1):
                    yield [choice] + rest

Usage:

for choice in permutations([['R'] * 2, ['B'] * 2], 4):
    print(choice)


['R', 'R', 'B', 'B']
['R', 'B', 'R', 'B']
['R', 'B', 'B', 'R']
['B', 'R', 'R', 'B']
['B', 'R', 'B', 'R']
['B', 'B', 'R', 'R']

Answer 2

Here's a "multi-combinations" implementation that doesn't require eliminating duplicate permutations. The first argument, n, is the list [Na, Nb, Nc, ...].

It is implemented as a recursive generator, so you can iterate through the combinations without having them all in memory at once. You say in a comment that Na + Nb + ... is typically around 20 but could be as high as 50 or 100. That means you almost certainly do not want to store all the combinations in memory. Consider an example with four "colors" where Na = Nb = Nc = Nd = 5. The number of combinations is choose(20, 5) * choose(15, 5) * choose(10, 5) = 11732745024, where choose(n, k) is the binomial coefficient. My computer has just 16 GB of RAM, so the storage required for that number of combinations would far exceed my computer's memory.

from itertools import combinations


def multicombinations(n, bins=None):
    if bins is None:
        bins = set(range(sum(n)))
    if len(n) == 0:
        yield []
    else:
        for c in combinations(bins, n[0]):
            for t in multicombinations(n[1:], bins - set(c)):
                yield [c] + t

It generates a list of tuples. That is, where your description of the first row is "First row is : R=(0,1) B=(2,3)", the first value generated by multicombinations([2, 2]) is [(0, 1), (2, 3)]. (This might not be the best format for the result, given the description of the weight calculation that you want to do next.)

Some examples:

In [74]: list(multicombinations([2, 2]))
Out[74]: 
[[(0, 1), (2, 3)],
 [(0, 2), (1, 3)],
 [(0, 3), (1, 2)],
 [(1, 2), (0, 3)],
 [(1, 3), (0, 2)],
 [(2, 3), (0, 1)]]

In [75]: list(multicombinations([3, 2]))
Out[75]: 
[[(0, 1, 2), (3, 4)],
 [(0, 1, 3), (2, 4)],
 [(0, 1, 4), (2, 3)],
 [(0, 2, 3), (1, 4)],
 [(0, 2, 4), (1, 3)],
 [(0, 3, 4), (1, 2)],
 [(1, 2, 3), (0, 4)],
 [(1, 2, 4), (0, 3)],
 [(1, 3, 4), (0, 2)],
 [(2, 3, 4), (0, 1)]]

In [76]: list(multicombinations([2, 3, 2]))
Out[76]: 
[[(0, 1), (2, 3, 4), (5, 6)],
 [(0, 1), (2, 3, 5), (4, 6)],
 [(0, 1), (2, 3, 6), (4, 5)],
 [(0, 1), (2, 4, 5), (3, 6)],
 [(0, 1), (2, 4, 6), (3, 5)],
 [(0, 1), (2, 5, 6), (3, 4)],
 [(0, 1), (3, 4, 5), (2, 6)],
 [(0, 1), (3, 4, 6), (2, 5)],
 [(0, 1), (3, 5, 6), (2, 4)],
 [(0, 1), (4, 5, 6), (2, 3)],
 [(0, 2), (1, 3, 4), (5, 6)],
 [(0, 2), (1, 3, 5), (4, 6)],
 [(0, 2), (1, 3, 6), (4, 5)],
 [(0, 2), (1, 4, 5), (3, 6)],
 [(0, 2), (1, 4, 6), (3, 5)],
 [(0, 2), (1, 5, 6), (3, 4)],
 [(0, 2), (3, 4, 5), (1, 6)],
 [(0, 2), (3, 4, 6), (1, 5)],
 [(0, 2), (3, 5, 6), (1, 4)],
 [(0, 2), (4, 5, 6), (1, 3)],
 [(0, 3), (1, 2, 4), (5, 6)],
 [(0, 3), (1, 2, 5), (4, 6)],
 [(0, 3), (1, 2, 6), (4, 5)],
 [(0, 3), (1, 4, 5), (2, 6)],
 [(0, 3), (1, 4, 6), (2, 5)],
 [(0, 3), (1, 5, 6), (2, 4)],
 [(0, 3), (2, 4, 5), (1, 6)],
 [(0, 3), (2, 4, 6), (1, 5)],
 [(0, 3), (2, 5, 6), (1, 4)],
 [(0, 3), (4, 5, 6), (1, 2)],
 [(0, 4), (1, 2, 3), (5, 6)],
 [(0, 4), (1, 2, 5), (3, 6)],
 [(0, 4), (1, 2, 6), (3, 5)],
 [(0, 4), (1, 3, 5), (2, 6)],
 [(0, 4), (1, 3, 6), (2, 5)],
 [(0, 4), (1, 5, 6), (2, 3)],
 [(0, 4), (2, 3, 5), (1, 6)],
 [(0, 4), (2, 3, 6), (1, 5)],
 [(0, 4), (2, 5, 6), (1, 3)],
 [(0, 4), (3, 5, 6), (1, 2)],
 [(0, 5), (1, 2, 3), (4, 6)],
 [(0, 5), (1, 2, 4), (3, 6)],
 [(0, 5), (1, 2, 6), (3, 4)],
 [(0, 5), (1, 3, 4), (2, 6)],
 [(0, 5), (1, 3, 6), (2, 4)],
 [(0, 5), (1, 4, 6), (2, 3)],
 [(0, 5), (2, 3, 4), (1, 6)],
 [(0, 5), (2, 3, 6), (1, 4)],
 [(0, 5), (2, 4, 6), (1, 3)],
 [(0, 5), (3, 4, 6), (1, 2)],
 [(0, 6), (1, 2, 3), (4, 5)],
 [(0, 6), (1, 2, 4), (3, 5)],
 [(0, 6), (1, 2, 5), (3, 4)],
 [(0, 6), (1, 3, 4), (2, 5)],
 [(0, 6), (1, 3, 5), (2, 4)],
 [(0, 6), (1, 4, 5), (2, 3)],
 [(0, 6), (2, 3, 4), (1, 5)],
 [(0, 6), (2, 3, 5), (1, 4)],
 [(0, 6), (2, 4, 5), (1, 3)],
 [(0, 6), (3, 4, 5), (1, 2)],
 [(1, 2), (0, 3, 4), (5, 6)],
 [(1, 2), (0, 3, 5), (4, 6)],
 [(1, 2), (0, 3, 6), (4, 5)],
 [(1, 2), (0, 4, 5), (3, 6)],
 [(1, 2), (0, 4, 6), (3, 5)],
 [(1, 2), (0, 5, 6), (3, 4)],
 [(1, 2), (3, 4, 5), (0, 6)],
 [(1, 2), (3, 4, 6), (0, 5)],
 [(1, 2), (3, 5, 6), (0, 4)],
 [(1, 2), (4, 5, 6), (0, 3)],
 [(1, 3), (0, 2, 4), (5, 6)],
 [(1, 3), (0, 2, 5), (4, 6)],
 [(1, 3), (0, 2, 6), (4, 5)],
 [(1, 3), (0, 4, 5), (2, 6)],
 [(1, 3), (0, 4, 6), (2, 5)],
 [(1, 3), (0, 5, 6), (2, 4)],
 [(1, 3), (2, 4, 5), (0, 6)],
 [(1, 3), (2, 4, 6), (0, 5)],
 [(1, 3), (2, 5, 6), (0, 4)],
 [(1, 3), (4, 5, 6), (0, 2)],
 [(1, 4), (0, 2, 3), (5, 6)],
 [(1, 4), (0, 2, 5), (3, 6)],
 [(1, 4), (0, 2, 6), (3, 5)],
 [(1, 4), (0, 3, 5), (2, 6)],
 [(1, 4), (0, 3, 6), (2, 5)],
 [(1, 4), (0, 5, 6), (2, 3)],
 [(1, 4), (2, 3, 5), (0, 6)],
 [(1, 4), (2, 3, 6), (0, 5)],
 [(1, 4), (2, 5, 6), (0, 3)],
 [(1, 4), (3, 5, 6), (0, 2)],
 [(1, 5), (0, 2, 3), (4, 6)],
 [(1, 5), (0, 2, 4), (3, 6)],
 [(1, 5), (0, 2, 6), (3, 4)],
 [(1, 5), (0, 3, 4), (2, 6)],
 [(1, 5), (0, 3, 6), (2, 4)],
 [(1, 5), (0, 4, 6), (2, 3)],
 [(1, 5), (2, 3, 4), (0, 6)],
 [(1, 5), (2, 3, 6), (0, 4)],
 [(1, 5), (2, 4, 6), (0, 3)],
 [(1, 5), (3, 4, 6), (0, 2)],
 [(1, 6), (0, 2, 3), (4, 5)],
 [(1, 6), (0, 2, 4), (3, 5)],
 [(1, 6), (0, 2, 5), (3, 4)],
 [(1, 6), (0, 3, 4), (2, 5)],
 [(1, 6), (0, 3, 5), (2, 4)],
 [(1, 6), (0, 4, 5), (2, 3)],
 [(1, 6), (2, 3, 4), (0, 5)],
 [(1, 6), (2, 3, 5), (0, 4)],
 [(1, 6), (2, 4, 5), (0, 3)],
 [(1, 6), (3, 4, 5), (0, 2)],
 [(2, 3), (0, 1, 4), (5, 6)],
 [(2, 3), (0, 1, 5), (4, 6)],
 [(2, 3), (0, 1, 6), (4, 5)],
 [(2, 3), (0, 4, 5), (1, 6)],
 [(2, 3), (0, 4, 6), (1, 5)],
 [(2, 3), (0, 5, 6), (1, 4)],
 [(2, 3), (1, 4, 5), (0, 6)],
 [(2, 3), (1, 4, 6), (0, 5)],
 [(2, 3), (1, 5, 6), (0, 4)],
 [(2, 3), (4, 5, 6), (0, 1)],
 [(2, 4), (0, 1, 3), (5, 6)],
 [(2, 4), (0, 1, 5), (3, 6)],
 [(2, 4), (0, 1, 6), (3, 5)],
 [(2, 4), (0, 3, 5), (1, 6)],
 [(2, 4), (0, 3, 6), (1, 5)],
 [(2, 4), (0, 5, 6), (1, 3)],
 [(2, 4), (1, 3, 5), (0, 6)],
 [(2, 4), (1, 3, 6), (0, 5)],
 [(2, 4), (1, 5, 6), (0, 3)],
 [(2, 4), (3, 5, 6), (0, 1)],
 [(2, 5), (0, 1, 3), (4, 6)],
 [(2, 5), (0, 1, 4), (3, 6)],
 [(2, 5), (0, 1, 6), (3, 4)],
 [(2, 5), (0, 3, 4), (1, 6)],
 [(2, 5), (0, 3, 6), (1, 4)],
 [(2, 5), (0, 4, 6), (1, 3)],
 [(2, 5), (1, 3, 4), (0, 6)],
 [(2, 5), (1, 3, 6), (0, 4)],
 [(2, 5), (1, 4, 6), (0, 3)],
 [(2, 5), (3, 4, 6), (0, 1)],
 [(2, 6), (0, 1, 3), (4, 5)],
 [(2, 6), (0, 1, 4), (3, 5)],
 [(2, 6), (0, 1, 5), (3, 4)],
 [(2, 6), (0, 3, 4), (1, 5)],
 [(2, 6), (0, 3, 5), (1, 4)],
 [(2, 6), (0, 4, 5), (1, 3)],
 [(2, 6), (1, 3, 4), (0, 5)],
 [(2, 6), (1, 3, 5), (0, 4)],
 [(2, 6), (1, 4, 5), (0, 3)],
 [(2, 6), (3, 4, 5), (0, 1)],
 [(3, 4), (0, 1, 2), (5, 6)],
 [(3, 4), (0, 1, 5), (2, 6)],
 [(3, 4), (0, 1, 6), (2, 5)],
 [(3, 4), (0, 2, 5), (1, 6)],
 [(3, 4), (0, 2, 6), (1, 5)],
 [(3, 4), (0, 5, 6), (1, 2)],
 [(3, 4), (1, 2, 5), (0, 6)],
 [(3, 4), (1, 2, 6), (0, 5)],
 [(3, 4), (1, 5, 6), (0, 2)],
 [(3, 4), (2, 5, 6), (0, 1)],
 [(3, 5), (0, 1, 2), (4, 6)],
 [(3, 5), (0, 1, 4), (2, 6)],
 [(3, 5), (0, 1, 6), (2, 4)],
 [(3, 5), (0, 2, 4), (1, 6)],
 [(3, 5), (0, 2, 6), (1, 4)],
 [(3, 5), (0, 4, 6), (1, 2)],
 [(3, 5), (1, 2, 4), (0, 6)],
 [(3, 5), (1, 2, 6), (0, 4)],
 [(3, 5), (1, 4, 6), (0, 2)],
 [(3, 5), (2, 4, 6), (0, 1)],
 [(3, 6), (0, 1, 2), (4, 5)],
 [(3, 6), (0, 1, 4), (2, 5)],
 [(3, 6), (0, 1, 5), (2, 4)],
 [(3, 6), (0, 2, 4), (1, 5)],
 [(3, 6), (0, 2, 5), (1, 4)],
 [(3, 6), (0, 4, 5), (1, 2)],
 [(3, 6), (1, 2, 4), (0, 5)],
 [(3, 6), (1, 2, 5), (0, 4)],
 [(3, 6), (1, 4, 5), (0, 2)],
 [(3, 6), (2, 4, 5), (0, 1)],
 [(4, 5), (0, 1, 2), (3, 6)],
 [(4, 5), (0, 1, 3), (2, 6)],
 [(4, 5), (0, 1, 6), (2, 3)],
 [(4, 5), (0, 2, 3), (1, 6)],
 [(4, 5), (0, 2, 6), (1, 3)],
 [(4, 5), (0, 3, 6), (1, 2)],
 [(4, 5), (1, 2, 3), (0, 6)],
 [(4, 5), (1, 2, 6), (0, 3)],
 [(4, 5), (1, 3, 6), (0, 2)],
 [(4, 5), (2, 3, 6), (0, 1)],
 [(4, 6), (0, 1, 2), (3, 5)],
 [(4, 6), (0, 1, 3), (2, 5)],
 [(4, 6), (0, 1, 5), (2, 3)],
 [(4, 6), (0, 2, 3), (1, 5)],
 [(4, 6), (0, 2, 5), (1, 3)],
 [(4, 6), (0, 3, 5), (1, 2)],
 [(4, 6), (1, 2, 3), (0, 5)],
 [(4, 6), (1, 2, 5), (0, 3)],
 [(4, 6), (1, 3, 5), (0, 2)],
 [(4, 6), (2, 3, 5), (0, 1)],
 [(5, 6), (0, 1, 2), (3, 4)],
 [(5, 6), (0, 1, 3), (2, 4)],
 [(5, 6), (0, 1, 4), (2, 3)],
 [(5, 6), (0, 2, 3), (1, 4)],
 [(5, 6), (0, 2, 4), (1, 3)],
 [(5, 6), (0, 3, 4), (1, 2)],
 [(5, 6), (1, 2, 3), (0, 4)],
 [(5, 6), (1, 2, 4), (0, 3)],
 [(5, 6), (1, 3, 4), (0, 2)],
 [(5, 6), (2, 3, 4), (0, 1)]]

Answer 3

You can solve that with itertools:

import itertools
last = None
result = []
for v in itertools.permutations(['A','A','B','B']):
    key = "".join(v)
    if key == last: continue
    last = key
    result.append(v)
print result

Answer 4

Generate the permutations and remove duplicates.

>>> import itertools
>>> Na = 2
>>> Nb = 2
>>> p = itertools.permutations(['R']*Na + ['B']*Nb)
>>> for perm in sorted(list(set(p)), reverse=True):
...     print perm
... 
('R', 'R', 'B', 'B')
('R', 'B', 'R', 'B')
('R', 'B', 'B', 'R')
('B', 'R', 'R', 'B')
('B', 'R', 'B', 'R')
('B', 'B', 'R', 'R')