Multi-latin squares

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dc.contributor.authorCavenagh, Nicholas J.
dc.contributor.authorHämäläinen, Carlo
dc.contributor.authorLefevre, James G.
dc.contributor.authorStones, Douglas S.
dc.date.accessioned2010-07-29T01:53:51Z
dc.date.available2010-07-29T01:53:51Z
dc.date.issued2010
dc.description.abstractA multi-latin square of order n and index k is an n×n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n≥2m, thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n≥k+2. We also show that for each n≥1, there exists some finite value g(n) such that for all k≥g(n), every k-latin square of order n is separable. We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders.en_NZ
dc.identifier.citationCavenagh, N.J., Hämäläinen, C., Lefevre, J.G. & Stones, D.S. (2010). Multi-latin squares. Discrete Mathematics, 311(13), 1164-1171.en_NZ
dc.identifier.doi10.1016/j.disc.2010.06.026en_NZ
dc.identifier.urihttps://hdl.handle.net/10289/4207
dc.language.isoen
dc.publisherElsevieren_NZ
dc.relation.isPartOfDiscrete Mathematicsen_NZ
dc.subjectlatin squareen_NZ
dc.subjectmulti-latin squareen_NZ
dc.subjectorthogonal arrayen_NZ
dc.subjectsemi-latin squareen_NZ
dc.subjectSOMAen_NZ
dc.subjectlatin parallelepipeden_NZ
dc.titleMulti-latin squaresen_NZ
dc.typeJournal Articleen_NZ
pubs.begin-page1164en_NZ
pubs.elements-id35184
pubs.end-page1171en_NZ
pubs.issue13en_NZ
pubs.volume311en_NZ
uow.identifier.article-no13en_NZ
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