Find all combinations of a given set of integers summing up to a given sum

Question

I am looking for an answer to the following problem.

Given a set of integers (no duplicates) and a sum, find all possible combinations of the set\'s elements summing

Answer 1

Here is a Haskell function that calculates the answer:

partitions 0 xs = [[]]
partitions _ [] = []
partitions n (xxs@(x:xs)) | n < 0 = []
                          | otherwise = (map (x:) (partitions (n-x) xxs)) ++ partitions n xs

Examples:

*Main>  partitions 1 [1]
[[1]]
*Main>  partitions 5 [1..5]
[[1,1,1,1,1],[1,1,1,2],[1,1,3],[1,2,2],[1,4],[2,3],[5]]
*Main> length $ partitions 10 [1..10]
42
*Main> length $ partitions 20 [1..20]
627
*Main> length $ partitions 40 [1..40]
37338
*Main> partitions 10 [1,2,4]
[[1,1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,2],[1,1,1,1,1,1,2,2],[1,1,1,1,1,1,4],[1,1,1,1,2,2,2],[1,1,1,1,2,4],[1,1,2,2,2,2],[1,1,2,2,4],[1,1,4,4],[2,2,2,2,2],[2,2,2,4],[2,4,4]]

Semi-live demo