Dominance Order on Signed Integer Partitions

In 1973 Brylawski introduced and studied in detail the dominance partial order on the set Par(m) of all integer partitions of a fixed positive integer m. As it is well known, the dominance order is one of the most important partial orders on the finite set Par(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, m can be an arbitrary integer and Par(m) becomes an infinite set. In this paper we extend the classical dominance order in this more general case. In particular, we consider the resulting lattice Par(m) as an infinite increasing union on n of a sequence of finite lattices O(m, n). The lattice O(m, n) can be considered a generalization of the Brylawski lattice. We study in detail the lattice structure of O(m, n).