Permutations $r_j$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized

TL;DR

This paper investigates the problem of finding permutations that maximize or minimize sums and products involving permutations, providing explicit solutions for certain cases and discussing computational challenges.

Contribution

It offers explicit solutions for specific values of n and k and explores the dual problem and variants of rearrangement inequalities.

Findings

Explicit solutions for certain n and k values

Discussion of computational issues in the general case

Analysis of a variant of the rearrangement inequality

Abstract

We consider the problem of finding the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized. While the set of permutations maximizing this value are easily determined, finding the set of permutations minimizing this value appears to be an open problem. We show values of $k$ and $n$ for which an explicit solution exists and comment on computational issues in determining the general problem. We also look at the dual problem of finding the permutations such that $\prod_{i=1}^n \sum_{j=1}^k r_j(i)$ is maximized or minimized. As part of this study we also look at a variant of a rearrangement inequality.