## Dominance order on signed integer partitions.(English)Zbl 1383.05020

Summary: T. Brylawski [Discrete Math. 6, 201–219 (1973; Zbl 0283.06003)] introduced and studied in detail the dominance partial order on the set $$\mathrm{Par}(m)$$ 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 $$\mathrm{Par}(m)$$. 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 $$\mathrm{Par}(m)$$ becomes an infinite set. In this paperwe extend the classical dominance order in this more general case. In particular, we consider the resulting lattice $$\mathrm{Par}(m)$$ 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 $$O(m, n)$$.

### MSC:

 05A17 Combinatorial aspects of partitions of integers 06A07 Combinatorics of partially ordered sets 11P81 Elementary theory of partitions 06B05 Structure theory of lattices

### Keywords:

number partitions; dominance order

Zbl 0283.06003
Full Text:

### References:

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