Next Composition of n into k parts - does anyone have a working algorithm?

What you are attempting to list is called an k-multicombination. The problem is often stated this way: given n indistinguishable balls and k boxes, list all possible ways to distribute all of the balls in the boxes. The number of such distributions is:

factorial(n + k - 1) / (factorial(k - 1) * factorial(n))

For further background, see Method 4 of the Twelve-Fold Way.

Here is the code to enumerate the distributions (C++):

string & ListMultisets(unsigned au4Boxes, unsigned au4Balls, string & strOut = string ( ), string strBuild = string ( ))
{
    unsigned au4;
    if (au4Boxes > 1) for (au4 = 0; au4 <= au4Balls; au4++)
    {
        stringstream ss;
        ss << strBuild << (strBuild.size() == 0 ? "" : ",") << au4Balls - au4;
        ListMultisets (au4Boxes - 1, au4, strOut, ss.str ( ));
    }
    else
    {
        stringstream ss;
        ss << "(" << strBuild << (strBuild.size() == 0 ? "" : ",") << au4Balls << ")\n";
        strOut += ss.str ( );
    }
    return strOut;
}



int main(int argc, char * [])
{    
    cout << endl << ListMultisets (3, 5) << endl;
    return 0;
}

Here is the output from the above program (5 balls distributed over three boxes):

(5,0,0)
(4,1,0)
(4,0,1)
(3,2,0)
(3,1,1)
(3,0,2)
(2,3,0)
(2,2,1)
(2,1,2)
(2,0,3)
(1,4,0)
(1,3,1)
(1,2,2)
(1,1,3)
(1,0,4)
(0,5,0)
(0,4,1)
(0,3,2)
(0,2,3)
(0,1,4)
(0,0,5)

Since it took me a bit of effort to read the intention of the other c++ solution here a translation to python (also as generator result instead of string):

def weak_compositions(boxes, balls, parent=tuple()):
  if boxes > 1:
    for i in xrange(balls + 1):
      for x in weak_compositions(boxes - 1, i, parent + (balls - i,)):
        yield x
  else:
    yield parent + (balls,)

test:

>>> for x in weak_compositions(3, 5): print x
(5, 0, 0)
(4, 1, 0)
(4, 0, 1)
(3, 2, 0)
...
(0, 1, 4)
(0, 0, 5)