  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.
