On properties of the generalized majorization

Abstract

In this paper, a complete solution of a problem involving generalized majorization of partitions is given: for two pairs of partitions (d, a) and (c, b) necessary and sufficient conditions for the existence of a partition g that is majorized by both pairs is determined. The obtained conditions are explicit, the solution is constructive and it uses novel techniques and indices. Although the problem is motivated by the applications in matrix pencil completions problems, all results are purely combinatorial and they give a new perspective on comparison of partitions.