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 A328873 Maximal size of a set of pairwise mutually orthogonal diagonal Latin squares of order n. 4
 1, 0, 0, 2, 2, 1, 4, 6, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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 existance 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(9) >= 6. - Eduard I. Vatutin, Oct 29 2019 a(n) <= A001438(n). - Max Alekseyev, Nov 08 2019 a(10) >= 2; a(11) >= 8; a(12) >= 2; a(13) >= 10; a(14) >= 2; a(15) >= 4. - Natalia Makarova, Sep 03 2020 Conjecture: a(9) = 6. - Natalia Makarova, Dec 24 2020 a(16) >= 14, a(17) >= 14, a(18) >= 2, a(19) >= 16, a(20) >= 2. - Natalia Makarova, Jan 08 2021 LINKS 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 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, Proving list (best known examples. Wikipedia, Clique problem. EXAMPLE 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. CROSSREFS Cf. A001438, A274806, A287695. Sequence in context: A193597 A191490 A061598 * A071946 A053495 A096747 Adjacent sequences:  A328870 A328871 A328872 * A328874 A328875 A328876 KEYWORD nonn,more,hard,changed AUTHOR Eduard I. Vatutin, Oct 29 2019 EXTENSIONS a(6) corrected by Max Alekseyev and Andrew Howroyd, Nov 08 2019 a(9) added by Eduard I. Vatutin, Feb 02 2021 STATUS approved

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Last modified April 11 19:32 EDT 2021. Contains 342888 sequences. (Running on oeis4.)