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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

Citations:

Zbl 0283.06003
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