Python Program to write an integer as a sum of r positive integers

Question

I want to create a python program which returns all possible ways of writing an integer (say n) as a sum of r positive integers.

My code is as follows:

def f(n,r):
cache = {}

l1 = []
for i in range(1,n):
    l1.append([i,n-i])
    
    
if r ==2 :
    cache[tuple([n,r])] = l1
    return l1

elif (n,r) in cache:
    return cache[(n,r)]
    
else:
    lr = []
    for i in range (1,n):
        for x in f(n-i,r-1):
            x.append(i)
            lr.append(x)
    cache[tuple([n,r])] = lr        
    return lr

It works well till n = 20 for any value of r but after that it's really slow, which is kinda expected because recurrence is slow. How can I achieve this without using recurrence?

Answer 1

A simple recursion with lru_cache :

>>> from functools import lru_cache
>>> @lru_cache(None)
... def f(n, r):
...     if r == 1:
...         return (n,),
...     return tuple((i,) + comb for i in range(1, n) for comb in f(n - i, r - 1))
...
>>> f(20, 3)
((1, 1, 18),
 (1, 2, 17),
 (1, 3, 16),
 ...
 (17, 1, 2),
 (17, 2, 1),
 (18, 1, 1))

Note: if we fix r to 10 and continue to increase n , the running time of the function will still gradually become unacceptable, because every time n increases by 1, the length of the obtained result will greatly increase, and the time spent on the concatenated of tuples (or lists) will also be more and more:

>>> {i: len(f(i, 10)) for i in range(10, 30)}
{10: 1,
 11: 10,
 12: 55,
 13: 220,
 14: 715,
 15: 2002,
 16: 5005,
 17: 11440,
 18: 24310,
 19: 48620,
 20: 92378,
 21: 167960,
 22: 293930,
 23: 497420,
 24: 817190,
 25: 1307504,
 26: 2042975,
 27: 3124550,
 28: 4686825,
 29: 6906900}

Answer 2

Generation in lexicographic order.

Note that any generation method won't be fast for large n,r values due to exponential grow of number of partitions.

from itertools import combinations
def parts(n,r):
    result = []
    for comb in combinations(range(n-1), r-1):
        parts = [comb[0]+1]+[comb[i]-comb[i-1] for i in range(1,r-1)]+[n-1-comb[-1]]
        #print(comb, parts)
        result.append(parts)
    return(result)

print(parts(6,3))

>>>[[1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 1, 3], [2, 2, 2], 
    [2, 3, 1], [3, 1, 2], [3, 2, 1], [4, 1, 1]]