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%I #85 May 21 2022 14:50:12
%S 1,2,3,4,1,6,7,8
%N Maximal number of mutually orthogonal Latin squares (or MOLS) of order n.
%C By convention, a(0) = a(1) = infinity.
%C Parker and others conjecture that a(10) = 2.
%C It is also known that a(11) = 10, a(12) >= 5.
%C It is known that a(n) >= 2 for all n > 6, disproving a conjecture by Euler that a(4k+2) = 1 for all k. - _Jeppe Stig Nielsen_, May 13 2020
%D CRC Handbook of Combinatorial Designs, 1996, pp. 113ff.
%D S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, Chapter 8.
%D E. T. Parker, Attempts for orthogonal latin 10-squares, Abstracts Amer. Math. Soc., Vol. 12 1991 #91T-05-27.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1997, p. 58.
%H Anonymous, <a href="http://web.archive.org/web/20071019111222/http://hypo.ge-dip.etat-ge.ch/www/math/gif/grecolatind.png">Order-10 Greco-Latin square</a>.
%H R. C. Bose and S. S. Shrikhande, <a href="http://www.pnas.org/cgi/reprint/45/5/734.pdf">On The Falsity Of Euler's Conjecture About The Non-Existence Of Two Orthogonal Latin Squares Of Order 4t+2</a>, Proc. Nat. Acad. Sci., 1959 45 (5) 734-737.
%H R. Bose, S. Shrikhande, and E. Parker, <a href="https://doi.org/10.4153/CJM-1960-016-5">Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture</a>, Canadian Journal of Mathematics, 12 (1960), 189-203.
%H C. J. Colbourn and J. H. Dinitz, <a href="http://www.emba.uvm.edu/~jdinitz/preprints/molssurv.pdf">Mutually Orthogonal Latin Squares: A Brief Survey of Constructions</a>, preprint, Journal of Statistical Planning and Inference, Volume 95, Issues 1-2, 1 May 2001, Pages 9-48.
%H M. Dettinger, <a href="https://web.archive.org/web/20090313020431/http://swiss2.whosting.ch/mdetting/eulersquare.html">Euler's Square</a>
%H David Joyner and Jon-Lark Kim, <a href="http://dx.doi.org/10.1007/978-0-8176-8256-9_3">Kittens, Mathematical Blackjack, and Combinatorial Codes</a>, Chapter 3 in Selected Unsolved Problems in Coding Theory, Applied and Numerical Harmonic Analysis, Springer, 2011, pp. 47-70, DOI: 10.1007/978-0-8176-8256-9_3.
%H Numberphile, <a href="https://www.youtube.com/watch?v=HuIrUeODtVQ">Euler squares</a>, YouTube video, 2020.
%H E. T. Parker, <a href="http://www.pnas.org/cgi/reprint/45/6/859.pdf">Orthogonal Latin Squares</a>, Proc. Nat. Acad. Sci., 1959 45 (6) 859-862.
%H E. Parker-Woodruff, <a href="https://web.archive.org/web/20070613155923/http://www2.msstate.edu/~eaddy/html/etparker.htm">Greco-Latin Squares Problem</a>
%H Tony Phillips, <a href="http://www.math.stonybrook.edu/~tony/whatsnew/column/latin-squaresII-0901/latinII3.html">Mutually Orthogonal Latin Squares (MOLS)</a>, Latin Squares in Practice and in Theory II.
%H N. Rao, <a href="http://bhavana.org.in/shrikhande-eulers-spoiler-turns-100/">Shrikhande, "Euler's Spoiler", Turns 100</a>, Bhāvanā, The mathematics magazine, Volume 1, Issue 4, 2017.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulersGraeco-RomanSquaresConjecture.html">Euler's Graeco-Roman Squares Conjecture</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Graeco-Latin_square">Graeco-Latin square</a>.
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%F a(n) <= n-1 for all n>1. - _Tom Edgar_, Apr 27 2015
%F a(p^k) = p^k-1 for all primes p and k>0. - _Tom Edgar_, Apr 27 2015
%F a(n) = A107431(n,n) - 2. - _Floris P. van Doorn_, Sep 10 2019
%Y Cf. A287695, A328873.
%K nonn,hard,more,nice
%O 2,2
%A _N. J. A. Sloane_
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