PlanetMath: Graeco-Latin squares

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Graeco-Latin squares

(Definition)

Let $A=(a_{ij})$ and $B=(b_{ij})$ be two $n\times n$ matrices. We define their join as the matrix whose th entry is the pair $(a_{ij},b_{ij})$ .

A Graeco-Latin square is then the join of two Latin squares.

The name comes from Euler's use of Greek and Latin letters to differentiate the entries on each array.

An example of Graeco-Latin square:

$\displaystyle \begin{pmatrix}a\alpha & b\beta & c\gamma & d\delta\\ d\gamma & c... ...a\gamma & d\beta & c\alpha\\ c\beta & d\alpha & a\delta & b\gamma \end{pmatrix}$

"Graeco-Latin squares" is owned by Mathprof. [ full author list (2) | owner history (2) ]

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Also defines:

join

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Cross-references: differentiate, Euler's, Latin squares, matrices
There are 6 references to this entry.

This is version 5 of Graeco-Latin squares, born on 2002-01-31, modified 2006-10-05.
Object id is 1625, canonical name is GraecoLatinSquares.
Accessed 3779 times total.

Classification:

AMS MSC:

05B15 (Combinatorics :: Designs and configurations :: Orthogonal arrays, Latin squares, Room squares)

Pending Errata and Addenda

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Discussion

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Graeco-Latin squares of order 7 and 8 by wpaulsen on 2003-03-05 10:52:49

Does anyone know whether there has been a complete search for all Graeco-Latin squares of order 7 or 8? (Not counting isomorphisms)
It is easy to verify that there is one square of order 3, one square of order 4, and two non-isomorphic Graeco-Latin squares of order 5.
Since it has been proven that there are no Graeco-Latin squares of order 6, the natural question is how many squares of order 7 or 8 there are.
Two Graeco-Latin squares are said to be isomorphic if one can be converted to the other through a series of operations: rearranging rows, rearranging columns, transposing the square, permuting the Greek letters, permuting the Latin letters, or exchanging the Greek and Latin lette

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