有没有办法递归地找到一组数字可以加起来为某个数字的不同方式？

Question

I want to be able to find all the different ways a set of numbers(x) can be summed into a certain value, y but I'm having trouble even getting the base case right.

For example:

If I have x = set (1,2,3,4,5) and I want to see how many different ways y = 5 can be summed up using numbers from x:

my recursive function would return 7 because:

'''
5
4+1 
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1
'''
def recur(x,y):
    if y == x:
        v += 1
    if y > x:
        v += 0
    else: 
         #call recursively

Answer 1

This does not use recursion, but itertools.combinations_with_replacement :

def all_combs(y, x=range(1, 5+1)):
    all_combs = []
    for i in range(1, y+1):
        combs = combinations_with_replacement(x, i)
        all_combs.extend([comb for comb in combs if sum(comb) == y])
    return all_combs

combs = all_combs(5)
# [(5,), (1, 4), (2, 3), (1, 1, 3), (1, 2, 2), (1, 1, 1, 2), (1, 1, 1, 1, 1)]
num_combs = len(combs) # 7