The following is a fundamental problem which we are not able to find a solution to in the literature. Given a vector $x \in R^n_{++}$ (positive numbers), denote by $x^\sigma$ an ordering of the elements in $x$ according to a permutation $\sigma \in S_n$. We wish to find a solution to the following optimization problem: \begin{equation*} \sigma^* \in \arg\max_{\sigma \in S_n} \sum_{i=1}^{n-m}\prod_{k=i}^{i+m} x^\sigma_i. \end{equation*} That is, what is the optimal ordering of the vector so that the sum of adjacent products is maximized. We conjecture that the optimal permutation does not depend on the numbers themselves and is ordered in a pendulum arrangement. That is, the largest value is in the middle, then the next values alternate from left to right until the smallest numbers reach the edges. That is, \begin{equation*} …,a(n−5),a(n−3),a(n−1),a(n),a(n−2),a(n−4),… \end{equation*} The proof for the case of $m=2$ can be found here.
Remark: We have already proved the following claim. An optimal permutation must have a form of a pyramid. That is, the optimal permutation is one in which the values monotonically increase up to some $j \in \{2, \ldots, n-1\}$ and monotonically decrease for all $i \geq j$.
The answer to this question can be found here: https://arxiv.org/pdf/2007.13232.pdf