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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 maximizedor minimized. While the set of permutations maximizing this value are easilydetermined, finding the set of permutations minimizing this value appears to bean open problem. We show values of $k$ and $n$ for which an explicit solutionexists 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 thisstudy we also look at a variant of a rearrangement inequality.

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