Suppose we 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 $\Pi_i \alpha_i$ can 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 the bound should be $n^2$, but I don't know I could prove it.