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 \(O(m, n)\).


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


Zbl 0283.06003
Full Text: DOI


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