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A001438 Maximal number of mutually orthogonal Latin squares (or MOLS) of order n. 1
1, 2, 3, 4, 1, 6, 7, 8 (list; graph; refs; listen; history; text; internal format)



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.

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


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.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 58 Penguin Books 1997.


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

Anonymous, Order-10 Greco-Latin square.

R. C. Bose & 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.

C. J. Colbourn & J. H. Dinitz, Mutually Orthogonal Latin Squares:A Brief Survey of Constructions

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.

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

E. Parker-Woodruff, Greco-Latin Squares Problem

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

Index entries for sequences related to Latin squares and rectangles


Sequence in context: A129708 A071518 A065338 * A105587 A049073 A076388

Adjacent sequences:  A001435 A001436 A001437 * A001439 A001440 A001441




N. J. A. Sloane



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Last modified November 25 18:43 EST 2015. Contains 264418 sequences.