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

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice
• changed

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

• Anonymous, Order-10 Greco-Latin square.
• Thomas Bloom, Problem 724, Erdős Problems.
• 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

Crossrefs

Cf. A287695, A328873.

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