×

zbMATH — the first resource for mathematics

Determinants of Latin squares of order 8. (English) Zbl 0876.05017
Summary: A Latin square is an \(n\times n\) array of \(n\) symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-\(n\) polynomial in \(n\) variables. Can two Latin squares \(L\), \(M\) have the same determinant, up to a renaming of the variables, apart from the obvious cases when \(L\) is obtained from \(M\) by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to be no if \(n\leq 7\); we show that it is yes for \(n=8\). The Latin squares for which this situation occurs have interesting special characteristics.

MSC:
05B15 Orthogonal arrays, Latin squares, Room squares
Software:
GAP
PDF BibTeX XML Cite
Full Text: DOI EMIS EuDML
References:
[1] Albert A. A., Trans. Amer. Math. Soc. 4 pp 507– (1943)
[2] Dénes J., Latin squares and their applications (1974) · Zbl 0283.05014
[3] Denes J., Latin squares: New developments in the theory and applications (1991)
[4] Ferguson M., ”The determinants of latin squares of order seven”, 1989
[5] Formanek E., Proc. Amer. Math. Soc. 112 pp 649– (1991)
[6] Frobenius G., Sitzungsber. Preuss. Akad. Wiss. Berlin Phys. Math. Kl. pp 985– (1896)
[7] Hawkins T., Arch. Hist. Exact Sci. 7 pp 142– (1971) · Zbl 0217.29903
[8] Hawkins T., Arch. Hist. Exact Sci. 12 pp 217– (1974) · Zbl 0397.01005
[9] Hoehnke H.-J., Bull. Amer. Math. Soc. (N.S.) 22 pp 243– (1992) · Zbl 0816.20010
[10] Johnson K. W., Algebraic, extremal and metric combinatorics pp 146– (1988)
[11] Johnson K. W., Math. Proc. Cambridge Phil. Soc. 109 pp 299– (1991) · Zbl 0742.20009
[12] Johnson K. W., Discrete Math 105 pp 111– (1992) · Zbl 0761.05019
[13] Johnson K. W., Group Theory pp 181– (1993)
[14] Kolesova G., J. Combin. Theory Ser. A 54 pp 143– (1990) · Zbl 0694.05015
[15] Muir T., A treatise on the theory of determinants (1960)
[16] Schönert M., GAP: Groups, algorithms, and programming (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.