More answers and solutions are always good, so thanks a lot for your effort!
The reason you're getting 27 combinations is that you're producing ordered tuples, whereas what I'm looking for is multisets (which are by definition unordered). I'll quote what I wrote in my post:
I'm trying to generate all multisets (bags) of a specific total "weight" (let's call it w), where each element comes from a given list (of numbers, in this case), and each list element may have multiplicity 0..w in each multiset. In other words, what I'm trying to generate is a list of w-tuples of elements of the given list — but unordered tuples rather than ordered ones.
What this means is that:
- Results that contain the same numbers a different amount of times are distinguished: "2,3,3" is not the same as "2,2,3".
- Results that are merely reordered are NOT distinguished: "2,3,3" is the same as "3,2,3" and "3,3,2".
Wikipedia has more on multisets: Multiset.
Like I said in my reply to tybalt89, the underlying problem I was trying to solve here¹ is related to a certain class of cellular automata with multiple states. I needed to generate all the possible combinations of states a certain subset of a given cell's immediate neighborhood could be in — but I was only interested in outer-totalistic CAs where the specific alignment of those neighboring cells didn't matter. Hence: it makes a difference whether of three cells I'm considering, one is in state 2 and two in state 3, or two in state 2 and one in state 3; but it doesn't make a difference specifically where in the center cell's neighborhood those neighboring cells are. Multisets / multicombinations were a natural choice for representing that.
TL;DR — thanks again, I appreciate all the good replies, tips, pointers and suggestions I got!
Footnotes:
- And which I have solved; it turns out that Algorithm::Combinatorics not only has a convenient function (combinations_with_repetitions) to generate just what I need, that function is also blindingly fast. See my reply to BrowserUk who also misunderstood (or didn't read carefully, one imagines) my question.
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