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      Multi-latin squares

      Cavenagh, Nicholas J.; Hämäläinen, Carlo; Lefevre, James G.; Stones, Douglas S.
      DOI
       10.1016/j.disc.2010.06.026
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      Cavenagh, N.J., Hämäläinen, C., Lefevre, J.G. & Stones, D.S. (2010). Multi-latin squares. Discrete Mathematics, 311(13), 1164-1171.
      Permanent Research Commons link: https://hdl.handle.net/10289/4207
      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.

      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.
      Date
      2010
      Type
      Journal Article
      Publisher
      Elsevier
      Collections
      • Computing and Mathematical Sciences Papers [1454]
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