Given a number of postcards (say 5) and a number of pigeon holes (say 3); how many ways are there to distribute the cards into the holes such that each hole contains at least 1?

For the 5/3 case above, the following possibilities exist:

3 1 1
2 2 1
2 1 2
1 2 2
1 1 3
1 3 1 // added per GrandFather's post below
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How to efficiently generate that sequence? The order of generation is immaterial.

Update: Need a better way

One way to do it, is to filter Algorithm::Combionatorics::variations_with_repetition() for the sum of values.

Ie. Generate the 27 variations:

And then filter on the sum() of the subsets to reduce it to the 6 I need:

1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
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Which doesn't seem too bad until you consider a realistic set, rather than my simple example.

For instance, with 12 postcards and 7 pigeon holes, there are 35,831,808 variations_with_repetition() each of which must be summed and compared in order to discard 99.9% to arrive at the 462 I need. That's horribly inefficient :(