I don't think you have a complete statement of the problem. Why, in the first example, are arrangements like 1 1 2 excluded? And you only accept the arrangements in a particular order. How do you define the order?
the data ascended by column,when the first column is the same, sort by the second column,...
Thanks - and I realize now that 1 1 2 is not excluded; I just wasn't looking carefully.
The best way to generate such a sequence is using depth-first search or recursion. By prohibiting the usage of IFs, all solutions are inefficient (because we can't cut the tree, and are forced to follow all branches).
For the 3 balls 2 boxes cases, shouldn't the following be part of the answer as well?
1 2 1
2 1 1
2 1 2
2 2 1
Indexed Probability Table
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