A001438 Maximal number of mutually orthogonal Latin squares (or MOLS) of order n.

1, 2, 3, 4, 1, 6, 7, 8

Offset

2,2

Comments

By convention, a(0) = a(1) = infinity.

Parker and others conjecture that a(10) = 2.

It is also known that a(11) = 10, a(12) >= 5.

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

References

CRC Handbook of Combinatorial Designs, 1996, pp. 113ff.

S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, Chapter 8.

E. T. Parker, Attempts for orthogonal latin 10-squares, Abstracts Amer. Math. Soc., Vol. 12 1991 #91T-05-27.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1997, p. 58.

Links

Table of n, a(n) for n=2..9.

Anonymous, Order-10 Greco-Latin square.

R. C. Bose and S. S. Shrikhande, On The Falsity Of Euler's Conjecture About The Non-Existence Of Two Orthogonal Latin Squares Of Order 4t+2, Proc. Nat. Acad. Sci., 1959 45 (5) 734-737.

R. Bose, S. Shrikhande, and E. Parker, Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture, Canadian Journal of Mathematics, 12 (1960), 189-203.

C. J. Colbourn and J. H. Dinitz, Mutually Orthogonal Latin Squares: A Brief Survey of Constructions, preprint, Journal of Statistical Planning and Inference, Volume 95, Issues 1-2, 1 May 2001, Pages 9-48.

M. Dettinger, Euler's Square

David Joyner and Jon-Lark Kim, Kittens, Mathematical Blackjack, and Combinatorial Codes, 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.

Numberphile, Euler squares, YouTube video, 2020.

E. T. Parker, Orthogonal Latin Squares, Proc. Nat. Acad. Sci., 1959 45 (6) 859-862.

E. Parker-Woodruff, Greco-Latin Squares Problem

Tony Phillips, Mutually Orthogonal Latin Squares (MOLS), Latin Squares in Practice and in Theory II.

N. Rao, Shrikhande, "Euler's Spoiler", Turns 100, Bhāvanā, The mathematics magazine, Volume 1, Issue 4, 2017.

Eric Weisstein's World of Mathematics, Euler's Graeco-Roman Squares Conjecture

Wikipedia, Graeco-Latin square.

Index entries for sequences related to Latin squares and rectangles

Formula

a(n) <= n-1 for all n>1. - Tom Edgar, Apr 27 2015

a(p^k) = p^k-1 for all primes p and k>0. - Tom Edgar, Apr 27 2015

a(n) = A107431(n,n) - 2. - Floris P. van Doorn, Sep 10 2019

Crossrefs

Cf. A287695, A328873.

Sequence in context: A316272 A294649 A350778 * A342410 A345937 A105587

Adjacent sequences: A001435 A001436 A001437 * A001439 A001440 A001441

Keyword

nonn,hard,more,nice

Author

N. J. A. Sloane

Status

approved