Monotone Optimal Multi-Partitions Using Schur Convexity With Respect To Partial Orders

In a (t,n,m,)-multi-partioning problem we partition t lists of nm ordered numbers into n sets, where each set contains m numbers from each list. The goal is to maximize some objective function that depends on the sum of the elements in each set and is called the partition function. We use the recently developed theory of majorization and Schur convexity with respect to partially ordered sets to study optimal solutions for the above problem. In particular, we construct a class of counterexamples to a recent conjecture of Du and Hwang which asserts that (classic) Schur convex functions can be characterized as the partition functions for (1,n,m)-multi-partitioning problems having optimal solutions that satisfy some monotonicity property.