Find all combinations of numbers from {x1,x2,...,xn} that sum of to sum S

Question

Given a list of some integers, I would like to find every combination that can be summed to some sum S. For example for the sum S=16, and the list of integers I={3,4,5}, I'd expect to get:

5,4,4,3 (=16) 5,5,3,3 (=16) 4,4,4,4 (=16)

How do I go about finding such lists for any given S, with any list of integers S?

Note, I can be a set of any integers they don't have to be sequential. Also note, I do understand this has to do with partitions, and have done reading to the best of my ability, but I have failed to fully understand how to make sense of this.

Any help will be highly appreciated!

I'm looking to understand how to construct such an algorithm. So far, I've been able to figure out how to calculate the amount of possible partitions, but I don't understand how figure every possible way to "fill" those partitions. — Michael Seltenreich, May 06 '20 at 14:01
You can just do it recursively. Take one of the integers $n$ in the set, consider the maximum number $m$ of the integer we can include, and run the algorithm with new sum $S-kn$ and integer set without $n$ for each $k$ from $0\to m$. — Rushabh Mehta, May 06 '20 at 14:04
I think I got it! Let me actually try to implement it, and see if I spoke too soon :) — Michael Seltenreich, May 06 '20 at 14:05
This is the same as the famous making change for a dollar problem. — lulu, May 06 '20 at 14:06
there is various of branching problems of Subset-Sum and knapsack with dynamic programming for them. i guess you should look alos for these problems, and you'll find what you look for (nobody's says those algorithms gonna be run fast though) — friedvir, May 06 '20 at 14:32
Don, I ended up successfully using your suggestion for algorithm. Many thanks! — Michael Seltenreich, May 06 '20 at 15:59

Find all combinations of numbers from {x1,x2,...,xn} that sum of to sum S

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