A287695 -id:A287695 - cp's OEIS frontend

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

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

Offset: 2

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

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.

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

Cf. A287695, A328873.

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

A328873 Maximal size of a set of pairwise mutually orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 2, 1, 4, 6, 6

Offset: 1

Eduard I. Vatutin, Oct 29 2019

From Andrew Howroyd, Nov 08 2019: (Start)
A diagonal Latin square of order n is an n X n array with every integer from 0 to n-1 in every row, every column, and both main diagonals.
Of course if even one example exists, then a(n) >= 1.
A274806 gives the number of diagonal Latin squares and A274806(6) is nonzero. This suggests that although it is not possible to have a pair of orthogonal diagonal Latin squares, a(6) should be 1 here. (End)
a(1) = 1 because there is only one (trivial) diagonal Latin square of order 1. It is orthogonal to itself, so if we allow the consideration of multiple copies of the same diagonal Latin square, we get a(1) = infinity instead.
From Eduard I. Vatutin, Mar 27 2021: (Start)
a(n) <= A287695(n) + 1.
a(p) >= A123565(p) = p-3 for all odd prime p due to existence of clique from cyclic MODLS of order p with at least A123565(p) items. It seems that for some orders p clique from cyclic MODLS can be extended by adding none cyclic DLS that are orthogonal to all cyclic DLS. (End)
a(n) <= A001438(n). - Max Alekseyev, Nov 08 2019
a(10) >= 2; a(11) >= 8; a(12) >= 4; a(13) >= 10; a(14) >= 2; a(15) >= 4. - Natalia Makarova, Sep 03 2020; updated May 30 2021
a(16) >= 14, a(17) >= 14, a(18) >= 2, a(19) >= 16, a(20) >= 2. - Natalia Makarova, Jan 08 2021

			Orthogonal pair of Diagonal Latin squares of order 18:
   1  5 15 16 17 18  2 14  4 13  3  7 12 10  8  6 11  9
   8  2  6 15 16 17 18  1  5 14  4 13 11  9  7 12 10  3
  14  9  3  7 15 16 17  2  6  1  5 12 10  8 13 11  4 18
  13  1 10  4  8 15 16  3  7  2  6 11  9 14 12  5 18 17
  12 14  2 11  5  9 15  4  8  3  7 10  1 13  6 18 17 16
  11 13  1  3 12  6 10  5  9  4  8  2 14  7 18 17 16 15
   3 12 14  2  4 13  7  6 10  5  9  1  8 18 17 16 15 11
   9 10 11 12 13 14  1 15 16 17 18  8  7  6  5  4  3  2
   6  7  8  9 10 11 12 18 17 16 15  5  4  3  2  1 14 13
   5  6  7  8  9 10 11 16 15 18 17  4  3  2  1 14 13 12
   7  8  9 10 11 12 13 17 18 15 16  6  5  4  3  2  1 14
   4 15 16 17 18  1  8 13  3 12  2 14  6 11  9  7  5 10
  15 16 17 18 14  7  9 12  2 11  1  3 13  5 10  8  6  4
  16 17 18 13  6  8  3 11  1 10 14 15  2 12  4  9  7  5
  17 18 12  5  7  2  4 10 14  9 13 16 15  1 11  3  8  6
  18 11  4  6  1  3  5  9 13  8 12 17 16 15 14 10  2  7
  10  3  5 14  2  4  6  8 12  7 11 18 17 16 15 13  9  1
   2  4 13  1  3  5 14  7 11  6 10  9 18 17 16 15 12  8
and
   1  8 14 13 12 11  3  9  6  5  7  4 15 16 17 18 10  2
   5  2  9  1 14 13 12 10  7  6  8 15 16 17 18 11  3  4
  15  6  3 10  2  1 14 11  8  7  9 16 17 18 12  4  5 13
  16 15  7  4 11  3  2 12  9  8 10 17 18 13  5  6 14  1
  17 16 15  8  5 12  4 13 10  9 11 18 14  6  7  1  2  3
  18 17 16 15  9  6 13 14 11 10 12  1  7  8  2  3  4  5
   2 18 17 16 15 10  7  1 12 11 13  8  9  3  4  5  6 14
  14  1  2  3  4  5  6 15 16 17 18 13 12 11 10  9  8  7
   4  5  6  7  8  9 10 17 18 15 16  3  2  1 14 13 12 11
  13 14  1  2  3  4  5 18 17 16 15 12 11 10  9  8  7  6
   3  4  5  6  7  8  9 16 15 18 17  2  1 14 13 12 11 10
   7 13 12 11 10  2  1  8  5  4  6 14  3 15 16 17 18  9
  12 11 10  9  1 14  8  7  4  3  5  6 13  2 15 16 17 18
  10  9  8 14 13  7 18  6  3  2  4 11  5 12  1 15 16 17
   8  7 13 12  6 18 17  5  2  1  3  9 10  4 11 14 15 16
   6 12 11  5 18 17 16  4  1 14  2  7  8  9  3 10 13 15
  11 10  4 18 17 16 15  3 14 13  1  5  6  7  8  2  9 12
   9  3 18 17 16 15 11  2 13 12 14 10  4  5  6  7  1  8
so a(18) >= 2.

R. J. R. Abel, Charles J. Colbourn, and Jeffrey H. Dinitz, Mutually Orthogonal Latin Squares (MOLS) [Note the first author, Julian Abel, has the initials R. J. R. A. - _N. J. A. Sloane_, Nov 05 2020]
B. Du, New Bounds For Pairwise Orthogonal Diagonal Latin Squares, Australasian Journal of Combinatorics 7 (1993), pp.87-99.
Natalia Makarova, MODLS of order 15
Natalia Makarova, Complete MOLS systems
Natalia Makarova, Orthogonal Diagonal Latin squares
Natalia Makarova, Mutually Orthogonal Diagonal Latin squares (MODLS) for orders 9 - 20
Natalia Makarova, MOLS and MODLS of order 12
E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian), Oct 29 2019.
Eduard I. Vatutin, On the falsity of Makarova's proof that a(9) = 6 (in Russian).
Eduard I. Vatutin, About the cliques from orthogonal diagonal Latin squares of order 9, brute force based proof that a(9) = 6 (in Russian).
E. I. Vatutin, M. O. Manzuk, V. S. Titov, S. E. Kochemazov, A. D. Belyshev, N. N. Nikitina, Orthogonality-based classification of diagonal latin squares of orders 1-8, High-performance computing systems and technologies. Vol. 3. No. 1. 2019. pp. 94-100. (in Russian).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, O. S. Zaikin, A. D. Belyshev, Cliques properties from diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2019). Tula, 2019. pp. 17-23. (in Russian).
Eduard I. Vatutin, About the A328873(N)-1 <= A287695(N) inequality between the maximum cardinality of clique and the maximum number of orthogonal normalized mates for one diagonal Latin square (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Wikipedia, Clique problem.
Index entries for sequences related to Latin squares and rectangles.

Cf. A001438, A274806, A287695.

a(6) corrected by Max Alekseyev and Andrew Howroyd, Nov 08 2019

a(9) added by Eduard I. Vatutin, Feb 02 2021

A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 3, 31, 99

Offset: 1

Eduard I. Vatutin, Jun 26 2021

a(n) <= A287695(n) + 1.
a(n) <= A287764(n).
a(10) >= 10. It seems that a(10) = 10 due to long computational experiments within the Gerasim@Home volunteer distributed computing project did not reveal the existence of diagonal Latin squares of order 10 with the number of orthogonal diagonal Latin squares different from {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.
a(11) >= 112, a(12) >= 5079. - Eduard I. Vatutin, Nov 02 2021, updated Jan 23 2023

			For n=7 the number of orthogonal diagonal Latin squares that a diagonal Latin square of order 7 may have is 0, 1, or 3. Since there are 3 distinct values, a(7)=3.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 10 (in Russian).
Eduard I. Vatutin, About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 11 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287695, A287764, A309344, A344105, A345370, A345760.

cp's OEIS Frontend

A330391 Number of main classes of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

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

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A328873 Maximal size of a set of pairwise mutually orthogonal diagonal Latin squares of order n.

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A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

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A382272 Maximum number of orthogonal diagonal Latin squares with the first row in ascending order that can be orthogonal to a given Brown's diagonal Latin square of order 2n.

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