Some constrained partitioning problems and majorization

We consider some constrained partitioning problems for a finite set of objects of different types. We look for partitions that are size- and type-similar, and, in addition, for a pair of such partitions that are ‘very different’ in a certain sense. The motivation stems from a problem involving the partitioning of a set of students into smaller groups. We give these problems precise mathematical formulations and investigate these problems using the notion of majorization. A special case of one of the problems leads to a result concerning the packing of matchings in a bipartite graph.