dc.contributor.author Cavenagh, Nicholas J. dc.contributor.author Hämäläinen, Carlo dc.contributor.author Lefevre, James G. dc.contributor.author Stones, Douglas S. dc.date.accessioned 2010-07-29T01:53:51Z dc.date.available 2010-07-29T01:53:51Z dc.date.issued 2010 dc.identifier.citation Cavenagh, N.J., Hämäläinen, C., Lefevre, J.G. & Stones, D.S. (2010). Multi-latin squares. Discrete Mathematics, 311(13), 1164-1171. en_NZ dc.identifier.uri https://hdl.handle.net/10289/4207 dc.description.abstract A 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. en_NZ 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. dc.language.iso en dc.publisher Elsevier en_NZ dc.subject latin square en_NZ dc.subject multi-latin square en_NZ dc.subject orthogonal array en_NZ dc.subject semi-latin square en_NZ dc.subject SOMA en_NZ dc.subject latin parallelepiped en_NZ dc.title Multi-latin squares en_NZ dc.type Journal Article en_NZ dc.identifier.doi 10.1016/j.disc.2010.06.026 en_NZ dc.relation.isPartOf Discrete Mathematics en_NZ pubs.begin-page 1164 en_NZ pubs.elements-id 35184 pubs.end-page 1171 en_NZ pubs.issue 13 en_NZ pubs.volume 311 en_NZ uow.identifier.article-no 13 en_NZ
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