

A001438


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


4




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, SpringerVerlag, NY, 1999, Chapter 8.
E. T. Parker, Attempts for orthogonal latin 10squares, Abstracts Amer. Math. Soc., Vol. 12 1991 #91T0527.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 58 Penguin Books 1997.


LINKS

Table of n, a(n) for n=2..9.
Anonymous, Order10 GrecoLatin square.
R. C. Bose & S. S. Shrikhande, On The Falsity Of Euler's Conjecture About The NonExistence Of Two Orthogonal Latin Squares Of Order 4t+2, Proc. Nat. Acad. Sci., 1959 45 (5) 734737.
R. Bose, S. Shrikhande, & 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), 189203.
C. J. Colbourn & J. H. Dinitz, Mutually Orthogonal Latin Squares: A Brief Survey of Constructions, preprint, Journal of Statistical Planning and Inference, Volume 95, Issues 12, 1 May 2001, Pages 948.
M. Dettinger, Euler's Square
David Joyner and JonLark Kim, Kittens, Mathematical Blackjack, and Combinatorial Codes, Chapter 3 in Selected Unsolved Problems in Coding Theory, Applied and Numerical Harmonic Analysis, Springer, 2011, pp. 4770, DOI: 10.1007/9780817682569_3.
Numberphile, Euler squares, YouTube video, 2020.
E. T. Parker, Orthogonal Latin Squares, Proc. Nat. Acad. Sci., 1959 45 (6) 859862.
E. ParkerWoodruff, GrecoLatin Squares Problem
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 GraecoRoman Squares Conjecture
Wikipedia, GraecoLatin square.
Index entries for sequences related to Latin squares and rectangles


FORMULA

a(n) <= n1 for all n>1.  Tom Edgar, Apr 27 2015
a(p^k) = p^k1 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: A065338 A316272 A294649 * A105587 A319676 A049073
Adjacent sequences: A001435 A001436 A001437 * A001439 A001440 A001441


KEYWORD

nonn,hard,more,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



