## Most Latin squares have many subsquares.(English)Zbl 0948.05014

This (in my opinion excellent and important) paper contains a wealth of results on subsquares of Latin squares. The attention is focused on subsquares of order 2 (intercalates). The methods and results are too technical to be reproduced here but some of the more easily stated corollaries include: (1) For arbitrary $$\varepsilon> 0$$, with probability approaching $$1$$ as $$n\to\infty$$, a random Latin squares of order $$n$$ contains at least $$n^{(3/2-\varepsilon)}$$ intercalates. (2) For arbitrary $$\varepsilon> 0$$, the probability of a random Latin square of order $$n$$ not containing any intercalates is $$O(\exp(-n^{(2-\varepsilon)})$$ as $$n\to\infty$$.
The results of computer enumeration for small order Latin square as well as a brief consideration of larger order subsquares are also included. As the authors suggest, a possible subtitle could be “…however, almost all of those subsquares are of order 2”.

### MSC:

 05B15 Orthogonal arrays, Latin squares, Room squares

### Keywords:

intercalates; subsquares; random Latin squares
Full Text:

### References:

 [1] Denniston, R. H.F., Remarks on Latin squares with no subsquares of order two, Utilitas Math., 13, 299-302 (1978) · Zbl 0379.05010 [2] Godsil, C. D.; McKay, B. D., Asymptotic enumeration of Latin rectangles, J. Combin. Theory Ser. B, 48, 19-44 (1990) · Zbl 0687.05010 [3] Heinrich, K.; Wallis, W. D., The maximum number of intercalates in a Latin square, Combinatorial Mathematics, VIII, Geelong, 1980. Combinatorial Mathematics, VIII, Geelong, 1980, Lecture Notes in Math., 884 (1981), Springer-Verlag: Springer-Verlag New York/Berlin, p. 221-233 · Zbl 0475.05014 [4] Jacobson, M. T.; Matthews, P., Generating uniformly distributed random Latin squares, J. Combin. Des., 4, 405-437 (1996) · Zbl 0913.05027 [5] Kotzig, A.; Lindner, C. C.; Rosa, A., Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math., 7, 287-294 (1975) · Zbl 0311.05013 [6] Kotzig, A.; Turgeon, J., On certain constructions for Latin squares with no Latin subsquares of order two, Discrete Math., 16, 263-270 (1976) · Zbl 0351.05016 [7] McKay, B. D.; Rogoyski, E., Electron. J. Combin., 2, N3 (1995) [8] McKay, B. D.; Wormald, N. C., Uniform generation of random Latin rectangles, Combin. Math. Combin. Comput., 9, 179-186 (1991) · Zbl 0751.05017 [9] McLeish, M., On the existence of Latin squares with no subsquares of order two, Utilitas Math., 8, 41-53 (1975) · Zbl 0342.05011 [10] Minc, H., Permanents. Permanents, Encyclopedia Math. Appl. (1978), Addison-Wesley: Addison-Wesley Reading · Zbl 0401.15005 [11] Pittenger, A. O., Mappings of Latin squares, Linear Algebra Appl., 261, 251-268 (1997) · Zbl 0876.05014 [12] Schrijver, A., Bounds on permanents and the number of 1-factors and 1-factorizations of bipartite graphs, London Math. Soc. Lecture Note Ser. (1983), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, p. 107-134 · Zbl 0529.05048 [13] van Lint, J. H.; Wilson, R. M., A Course in Combinatorics (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0769.05001
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