Find unique sums to add to total (using only positive integers above 0)

In the long run this will be used to find groups of words that spell another. (For crosswords)

But for now I just want to know some kind of generalized algorithm to calc the ways to get to a number using sums.

So lets say I have a 6 letter word, the only possible way to "create" that word from sub words would be words of length 5,4,3,2,1 (and I guess the word and word anagrams of that word):

6 = 6 
6 = 5 + 1
6 = 4 + 2
6 = 4 + 1 + 1
6 = 3 + 3
6 = 3 + 2 + 1
6 = 3 + 1 + 1 + 1
6 = 2 + 2 + 2
6 = 2 + 2 + 1 + 1
6 = 2 + 1 + 1 + 1 + 1
6 = 1 + 1 + 1 + 1 + 1 + 1

So I want code to generate me this in an array in python, so I can go look for words of combinations of lengths. I may want to exclude 1 letter words & same length words, but since there is just a max of 26 (and in English only I and A, and maybe if you include slang U) I don’t think it’ll make it too complex.

This doesn’t need to be too optimized since usually like 15 is the highest I will go (I might even just type it all out and then I have it pre-solved, but it’s going to be a fair amount for 15.

>Solution :

Here is a recursive solution to your question. We basically check all possible sum combinations until we find a valid one, then add it to our list of sums.

def find_sums(n):
    sums = []
    all_sums = []
    find_sums_helper(n, n, sums, all_sums)
    return all_sums

def find_sums_helper(n, n_max, sums, all_sums):
    if n == 0:
        all_sums.append(sums)
        return
    
    for i in reversed(range(1, min(n, n_max) + 1)):
        find_sums_helper(n - i, i, sums + [i], all_sums)

print(find_sums(6))

Output:

[[6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 2, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]]