Number of ways n can be written as sum of at least two positive integers

Question

I found a solution in Python for this problem, but do not understand it. The problem is how many ways an integer n can be written as the sum of at least two positive integers. For example, take n = 5. The number 5 can be written as

4 + 1
3 + 2
3 + 1 + 1
2 + 2 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1

Here's a solution for given $n$.

# zero-based array (first index is 0)
ways = [1, 0, ..., 0] (n zeroes)

for i in 1, ..., n-1:
    for j in i, ..., n:
        ways[j] = ways[j] + ways[j-i]

print ways[n]

This solution is elegant and efficient, but unfortunately, I do not understand the logic. Can someone please explain the logic of this solution to the problem? Is there a way to make this algorithm easy to understand?

Answer 1

Let us denote the array ways after $t$ iterations of the outer loop by $w_t$. The recurrence implemented by the code is $$ w_0(n) = \begin{cases} 1 & \text{if } n = 0, \\ 0 & \text{if } n > 0. \end{cases} \\ w_t(n) = \begin{cases} w_{t-1}(n) & \text{if } n < t, \\ w_{t-1}(n) + w_t(n-t) & \text{if } n \geq t. \end{cases} $$ You can prove by induction that $w_t(n)$ is the number of representations of $n$ as a sum of natural numbers between $1$ and $t$ (without regard to order). (This is also the number of representations of $n$ as a sum of at most $t$ natural numbers.)

The code returns $w_{n-1}(n)$, which is the number of representations of $n$ as an arbitrary sum of natural numbers, other than the representation $n$.

As an aside, if you allow the trivial representation then you get the partition function. Using Rademacher's asymptotic formula, the partition function can be computed in time $\tilde O(\sqrt{n})$, much faster than your $\Theta(n^2)$ algorithm.