Suppose I have $n$ integers (both negative and positive) and I get all combinations of $k$ elements with repetition $((n, k)) = (n + k-1, k)$
My question is: what is the maximum number of combinations, that the sum could be the same? That is, what is the maximum number of combinations with the same amount, I can get by carefully selecting the $n$ numbers. Assuming that two combinations whose elements were the same but in a different order, are the same combination.
I've been testing with 4 numbers selected such that an attempt to obtain the maximum number of combinations with the same sum. I noticed that either combining of 3 on 3 with repetition, 2 on 2 with repetition or 5 on 5, in all cases, the maximum number of combinations with the same sum, was always 3. I have never managed to fix the 4 numbers to obtain a larger number of combinations with the same amount.
If this were a general rule, then the maximum number of combinations with repetition that sum the same for $n$ elements, is always $n-1$, regardless of the value of $k$. Is it true?
Does anyone know where there is a general proof? Or a formula?