2
$\begingroup$

This is a follow-up to my previous MO question:

A discrete optimization problem related to the AM-GM inequality

Let $n,k$ be integers such that $1\le k\le n$. Define the quantity $$ P(n,k):=\max\ a_1\cdots a_k\ , $$ where the maximum is taken over $k$-tuples of integers $(a_1,\ldots,a_k)$ which are $\ge 1$ and which sum up to $n$.

One can write the explicit formula $$ P(n,k)=\left(\left\lfloor\frac{n}{k}\right\rfloor\right)^{k-n+\left\lfloor\frac{n}{k}\right\rfloor} \left(\left\lfloor\frac{n}{k}\right\rfloor+1\right)^{n-k\left\lfloor\frac{n}{k}\right\rfloor}\ . $$

I would like to learn as much as possible about what is known in the literature regarding the quantity $P(n,k)$. For instance, in the other MO question, a connection to the Turán graph was mentioned. Any references, which study $P(n,k)$ or where that quantity plays a significant role, would be appreciated. I tried to do a search on the web, but I mostly got links to finance articles about portfolio maximization which did not quite suit my needs.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
+50
$\begingroup$

If you maximize $P(n,k)$ over $k$ you obtain the OEIS sequence A000792, which appears in a variety of contexts (maximal size of an Abelian subgroup of the symmetric group $S_n$, maximum number of maximal cliques possible in a graph with $n$ vertices, largest number of complexity $n$, ...)

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ Thank you! Looking at the OEIS page opens up a whole chunk of the related literature I was not aware of. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 21, 2023 at 15:21
  • $\begingroup$ Just posted an article related to this question: arxiv.org/abs/2309.07358 $\endgroup$
    – Abdelmalek Abdesselam
    Commented Sep 18, 2023 at 17:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .