Dominance order on signed integer partitions
Cinzia Bisi, Giampiero Chiaselotti, Tommaso Gentile, Paolo Antonio Oliverio · Advances in Geometry · 2017
Abstract In 1973 Brylawski introduced and studied in detail the dominance partial order on the setPar(m) of all integer partitions of a fixed positive integerm. As it is well known, the dominance order is one of the most important partial orders on the finite setPar(m). Therefore it is very natural to ask how it changes if we allow the summands of an integer partition to take also negative values. In such a case,mcan be an arbitrary integer andPar(m) becomes an infinite set. In this paperwe extend the classical dominance order in this more general case. In particular, we consider the resulting latticePar(m) as an infinite increasing union onnof a sequence of finite latticesO(m,n). The latticeO(m,n) can be considered a generalization of the Brylawski lattice. We study in detail the lattice structure ofO(m,n).