Algebra and Topology Seminar

4pm Tuesday 19 February 2002, Moran G008

Wendy Myrvold
University of Victoria

Counting Small Latin Squares

A Latin square is an n by n array on n symbols where each symbol appears exactly once in each row and column. Two Latin squares are isotopic if one can be transformed to the other by permuting rows, columns, and/or symbols. Being in the same main class means that in addition, the roles of rows, columns and/or symbols can be interchanged. A third form of equivalence is isomorphism which will be defined in the talk. By using the novel technique of generating only the Latin squares having non-trivial symmetry group, we are able to count the number of isotopy, main, and isomorphism classes up to n = 9 and there is hope the results can be extended to n = 10.

An outstanding open question is whether there exists an orthogonal triple of Latin squares of order 10. The squares of order 10 which have been generated so far have been tested to see if they belong to some orthogonal triple. From these computations, we know that any square in such a triple must have isotopy group of order 2^k for some integer k >= 0, as all other squares have been tested. This is joint work with Brendan McKay.




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