McKay, Brendan D.; Meynert, Alison; Myrvold, Wendy Small Latin squares, quasigroups, and loops. (English) Zbl 1112.05018 J. Comb. Des. 15, No. 2, 98-119 (2007). Summary: We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 [G. Kolesova, C.W.H. Lam and L. Thiel, J. Comb. Theory, Ser. A 54, No. 1, 143–148 (1990; Zbl 0694.05015)], quasigroups of order 6 [C. G. Bower, Private communication (2000)], and loops of order 7 [L. J. Brant and G. L. Mullen, Util. Math. 27, 261–263 (1985; Zbl 0549.05011)]. The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 2001). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. Cited in 71 Documents MSC: 05B15 Orthogonal arrays, Latin squares, Room squares Keywords:isotopy; main class; orthogonal Citations:Zbl 0694.05015; Zbl 0549.05011 Software:nauty PDFBibTeX XMLCite \textit{B. D. McKay} et al., J. Comb. Des. 15, No. 2, 98--119 (2007; Zbl 1112.05018) Full Text: DOI Online Encyclopedia of Integer Sequences: Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element. Number of 1-factorizations of K_{n,n}. Number of Latin squares of order n; or labeled quasigroups. Number of species (or ”main classes” or ”paratopy classes”) of Latin squares of order n. Number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n. Number of loops (quasigroups with an identity element) of order n. Number of quasigroups of order n. Number of loops of order n that are not groups. References: [1] Acketa, Zb Rad Prirod-Mat Fak Ser Mat 25 pp 141– (1995) [2] Albert, Trans Amer Math Soc 55 pp 401– (1944) [3] Arlazarov, Algorithmic investigations in combinatoric (Moscow, Nauka, 1978) pp 129– [4] Bammel, Discrete Math 11 pp 93– (1975) [5] Bose, Proc Nat Acad Sci USA 45 pp 734– (1959) [6] Bose, Can J Math 12 pp 189– (1960) · Zbl 0093.31905 · doi:10.4153/CJM-1960-016-5 [7] Private communication (2000). [8] Brant, Utilitas Math 27 pp 261– (1985) [9] Brouwer, J Statist Plann Inference 10 pp 203– (1984) [10] Brown, J Combin Theory 5 pp 177– (1968) [11] Brown, J Combin Theory Ser A 12 pp 316– (1972) [12] Brown, Congr Numer 36 pp 143– (1982) [13] Brown, Congr Numer 43 pp 201– (1984) [14] Brown, Algebras Groups Geom 2 pp 258– (1985) [15] Brown, Ars Combin 35 pp 125– (1993) [16] Cayley, Oxford Camb Dublin Messenger Math 19 pp 85– (1890) [17] and (Editors), The CRC Handbook of Combinatorial Designs. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton, FL, 1996, 753. · doi:10.1201/9781420049954 [18] Colbourn, J Statist Plann Inference 95 pp 9– (2001) [19] Cruse, J Comb Theory Ser A 16 pp 18– (1974) [20] and , Latin Squares and their Applications, Academic Press, New York-London, 1974. · Zbl 0283.05014 [21] Euler, Verhandelingen/uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen 9 pp 85– (1782) [22] Faradžev, Colloq. Internat. CNRS pp 131– (1978) [23] Fisher, Proc Cambridge Philos Soc 30 pp 492– (1934) [24] Frolov, J de Math spéc pp 8– (1890) [25] Private communication (2001). [26] Jacob, Proc London Math Soc 31 pp 329– (1930) [27] Jacobson, J Combin Des 4 pp 405– (1996) [28] Kolesova, J Combin Theory Ser A 54 pp 143– (1990) [29] Lam, Can J Math 41 pp 1117– (1989) · Zbl 0691.51003 · doi:10.4153/CJM-1989-049-4 [30] Combinatory Analysis, Cambridge University Press, Cambridge, 1915. · JFM 46.0118.07 [31] Maenhaut, J Combin Des 12 pp 12– (2004) [32] nauty graph isomorphic software, available at http://cs.anu.edu.au/bdm/nauty. [33] McKay, J Algorithms 26 pp 306– (1998) [34] McKay, Electron J Combin 2 pp 4– (1995) [35] McKay, Ann Combin 9 pp 335– (2005) [36] Myrvold, J Combin Math Combin Comput 29 pp 95– (1999) [37] Nazarok, Akad. Nauk Ukrain. SSR Inst Mat, Kiev pp 89– (1991) [38] Neumann, Math Sci 4 pp 133– (1979) [39] Norton, Ann Eugenics 9 pp 269– (1939) · Zbl 0022.11102 · doi:10.1111/j.1469-1809.1939.tb02214.x [40] Parker, Proc Nat Acad Sci USA 45 pp 859– (1959) [41] Parker, Proc Sympos Appl Math Vol XV, (Amer Math Soc, 1963) pp 73– [42] Parker, J Combin Theory Ser A 19 pp 243– (1975) [43] Preece, J Royal Stat Soc Series B (Meth) 28 pp 118– (1966) [44] QSCGZ (pseudonym), Anonymous electronic posting to ”loopforum”, October 2001. http://groups.yahoo.com/group/loopforum/ [45] Read, Annals Discrete Math 2 pp 107– (1978) [46] Enumération des carrés latins. Application au 7ème ordre. Conjectures pour les ordres supérieurs, privately published, Marseille, 1948, 8 p. [47] Sade, Ann Math Stat 22 pp 306– (1951) [48] Sade, Tables, Univ Lisboa Revista Fac Ci A 13 pp 149– (1970/71) [49] Saxena, J Indian Soc Agric Statistics 3 pp 24– (1951) [50] Schönhardt, J Reine Angew Math 163 pp 183– (1930) [51] Shao, Discrete Math 110 pp 293– (1992) [52] Shrikhande, Sankhyā Ser A 23 pp 115– (1961) [53] Tarry, Ass Franç Paris 29 pp 170– (1900) [54] Wanless, Eur J Combin 22 pp 1009– (2001) [55] Wells, J Combin Theory 3 pp 98– (1967) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.