Sorry if i ask this question. probably it's already answered somewhere else but i didn't find it.
Suppose to have a natural number $n \ge 0$.
Given natural numbers $\alpha_1,\ldots,\alpha_k$ such that
- $k\le n$
- $\sum_i \alpha_i = n$
what is the maximum value that $\Pi_i \alpha_i$ could take?
I'm quite sure that there is a theorem telling me the result but i cannot find it. For sure an upper bound is $n^k$ but i'm searching for a real upper bound. I'm pretty sure that upper bound should be $n^2$ but i don't know i could prove it.