Count the number of ways of putting balls into bins

Question

In this task you are given an odd number of white balls and the same number of black balls. The task is to count all the ways of putting the balls into bins so that in each bin there is an odd number of each color.

For example, say we have 3 white balls. The different ways are:

(wwwbbb)
(wb)(wb)(wb)

for the two different possibilities.

If we have 5 white balls the different ways are:

(wwwwwbbbbb)
(wwwbbb)(wb)(wb)
(wwwb)(wbbb)(wb)
(wb)(wb)(wb)(wb)(wb)

You can take the input, which is a single integer, in any way you like. The output is just a single integer.

Your code must be fast enough so that you have seen it complete for 11 white balls.

You can use any language or library that you like.

Answer 1

Python 3, 108 bytes

C=lambda l,r,o=():((l,r)>=o)*l*r%2+sum(C(l-x,r-y,(x,y))for x in range(1,l,2)for y in range(1,r,2)if(x,y)>=o)

Recursively enumerates all sets, making sure to not get duplicates by always generating the sets in order. Reasonably fast when memoized using C = functoools.lru_cache(None)(C), but this is not necessary for n = 11.

Call C(num_white, num_black) to get your result. First couple of n:

1: 1
3: 2
5: 4
7: 12
9: 32
11: 85
13: 217
15: 539
17: 1316
19: 3146
21: 7374

To generate the results:

def odd_parts(l, r, o=()):
    if l % 2 == r % 2 == 1 and (l, r) >= o:
        yield [(l, r)]

    for nl in range(1, l, 2):
        for nr in range(1, r, 2):
            if (nl, nr) < o: continue
            for t in odd_parts(l - nl, r - nr, (nl, nr)):
                yield [(nl, nr)] + t

E.g. for (7, 7):

[(7, 7)]
[(1, 1), (1, 1), (5, 5)]
[(1, 1), (1, 1), (1, 1), (1, 1), (3, 3)]
[(1, 1), (1, 1), (1, 1), (1, 1), (1, 1), (1, 1), (1, 1)]
[(1, 1), (1, 1), (1, 1), (1, 3), (3, 1)]
[(1, 1), (1, 3), (5, 3)]
[(1, 1), (1, 5), (5, 1)]
[(1, 1), (3, 1), (3, 5)]
[(1, 1), (3, 3), (3, 3)]
[(1, 3), (1, 3), (5, 1)]
[(1, 3), (3, 1), (3, 3)]
[(1, 5), (3, 1), (3, 1)]

Answer 2

Pari/GP, 81 bytes

p=polcoeff;f(n)=p(p(prod(i=1,n,prod(j=1,n,1+(valuation(i/j,2)==0)*x^i*y^j)),n),n)

For more efficiency, replace 1+ with 1+O(x^(n+1))+O(y^(n+1))+ (the first O term alone already helps a lot).

Try it online! (earlier 86 byte version with a pair of unneeded parens and without the p= abbreviation)

Old version, 90 bytes

f(n)=polcoeff(polcoeff(taylor(1/prod(i=0,n,prod(j=0,n,1-x^(2*i+1)*y^(2*j+1))),x,n+1),n),n)

Computing f(11) needs a bigger stack size, the error message will tell you how to increase it. It's more efficient (but less golfy) to replace the two n that appear as second arguments to prod with (n-1)/2.

Answer 3

Python 3, 180 172 bytes

def f(n):
 r=range;N=n+1;a=[N*[0]for _ in r(N)];R=r(1,N,2);a[0][0]=1
 for i in R:
  for j in R:
   for k in r(N-i):
    for l in r(N-j):a[k+i][l+j]+=a[k][l]
 return a[n][n]

Try it online!

Straightforward implementation of the generating function. Long but (somewhat) efficient. O(n⁴) time, O(n²) memory.

The resulting array a contains all results of all sizes up to n, although only a[n][n] is returned.

Answer 4

Python 2, 168 181 bytes

from itertools import*
r,p=range,product
def f(n):
 a,R=eval(`[[0]*n]*n`),r(1,n,2);a[0][0]=1
 for i,j in p(R,R):
  for k,l in p(r(n-i),r(n-j)):a[k+i][l+j]+=a[k][l]
 return a[-1][-1]

Try it online!

• Based on user202729's answer.
• Edit: Made it an independent function.
• Saved 2 bytes thanks to user202729.