How do I generate integer partitions?

Question

I have a list of numbers like 1,2,3 and I want to find all the combination patterns that sum up to a particular number like 5. For example:

Sum=5
Numbers:1,2         


        

Answer 1

I would do it recursively starting with the highest numbers first. then, each time in start with the highest level and go in as many levels as numbers. As soon as the cumulative level exceeds your value, drop down to the next number. If still too large (or small), immediately return back one level and decrease THAT to the next number down, then to the next deeper level starting at the top again..