In how many ways can N be written as a sum of positive numbers less or equal to K?

Question

I'm trying to solve an informatics problem, and I need to know if there is a formula that can be applied to solve the problem.

Any help appreciated!

Edit: This question was flagged as a possible duplicate of some other question. I want to clarify that the number of numbers added is arbitrary, though the numbers added must be less or equal to K.

If you consider $3 = 2+1$ and $3 = 1+2$ to be the same way of writing $3$ as a sum of positive numbers, then there is very little in the way of formulas, even without the size restriction. The relevant buzzword is partition, and you may see this wikipedia article on that problem. — Arthur, Aug 08 '18 at 11:29
Possible duplicate of Number of ways to write n as a sum of k nonnegative integers — , Aug 08 '18 at 11:30
There is no simple formula, but there is a simple algorithm that the computer can use to do the calculation. Would that be helpful? — MJD, Aug 08 '18 at 11:30
Okay, and do you want to consider $4 = 2 +1+1$ as being the same way as $4 = 1 + 2 + 1$ and $4 = 1+1+2$, or are those three different ways? — MJD, Aug 08 '18 at 11:54

score 2 · Answer 1 · edited Aug 09 '18 at 09:32

This function will calculate the answer:

number_of_partitions(N, max):
    if      N = 0:   return 1
    else if N < 0:    return 0
    else:
        total ← 0
        for i in 1 … max do:
            total ← total + number_of_partitions(N-i, max)
        return total

For example, number_of_partitions(4,2) returns $5$, because it is counting

$$\begin{align} 4 &= 1+1+1+1\\ 4 &= 2 + 1 +1 \\ 4 &= 1 + 2 + 1 \\ 4 &= 1 + 1 + 2 \\ 4 &= 2 + 2 \end{align}$$

but not $$\begin{align}4 & = 3 +1 \\ 4& = 1+3\\4 & = 4.\end{align}$$

How does this work? The recursion is the important part. We are going to take the original number $N$ and split off parts one at a time. The part we split off can be any size $i$ from $1$ up to $max$. Then we count how many ways there are to partition the remainder $N-i$ and add up the results for each $i$.

The only subtle point is in the base case for the recursion. number_of_partitions(0, max) is always 1, because there is exactly one partition of $N=0$, namely the empty partition.

This counting considers $1+1+2$ to be distinct from $2+1+1$. If you want them to be the same, do the recursion with total ← total + number_of_partitions(N-i, i) instead, and it will only count the ways which are sorted largest-to-smallest.

In how many ways can N be written as a sum of positive numbers less or equal to K?

1 Answers1