A001438 -id:A001438 - cp's OEIS frontend

A287695 Maximum number of diagonal Latin squares with the first row in ascending order that can be orthogonal to a given diagonal Latin square of order n.

1, 0, 0, 1, 1, 0, 3, 824, 614

Offset: 1

Eduard I. Vatutin, May 30 2017

A Latin square is normalized if in the first row elements come in increasing order. Any diagonal Latin square orthogonal to a given one can be normalized by renaming its elements (which does not break diagonality and orthogonality). - Max Alekseyev, Dec 07 2019
For all orders n>3 there are diagonal Latin squares without orthogonal mates (also known as bachelor squares), so the minimum number of diagonal Latin squares that can be orthogonal to the same diagonal Latin square is zero. For order n=1 the single square is orthogonal to itself. For n=2 and n=3 diagonal Latin squares do not exist (see A274171). For n=6 orthogonal diagonal Latin squares do not exist (see A305571), so a(6)=0. - Eduard I. Vatutin, May 03 2021
a(n) >= A328873(n) - 1. - Eduard I. Vatutin, Mar 29 2021
a(10) >= 10 (Updated). - Eduard I. Vatutin, Apr 27 2018
a(11) >= 32462. - Eduard I. Vatutin from T. Brada, Mar 11 2021
a(12) >= 3855983322. The result belongs to DLS, which has 30192 diagonal transversals. Calculations performed by a volunteer. - Natalia Makarova, Tomáš Brada, Nov 11 2021
a(13) >= 248703. - Natalia Makarova, Tomáš Brada, Apr 29 2021
a(14) >= 307662. - Natalia Makarova, Alex Chernov, Harry White, May 21 2021
a(16) >= 1658880, a(17) >= 2453352, a(18) >= 96, a(19) >= 1383, a(20) >= 995328, a(21) >= 995328, a(22) >= 432000, a(23) >= 525, a(24) >= 345600, a(25) >= 345600, a(26) >= 48, a(27) >= 345600, a(28) >= 663552, a(29) >= 663552, a(30) >= 40320. For values up to a(100), see the specified link "New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square". - Natalia Makarova, Alex Chernov, Harry White, Dec 06 2021

			From _Eduard I. Vatutin_, Mar 29 2021: (Start)
One of the best existing diagonal Latin squares of order 7
  0 1 2 3 4 5 6
  2 3 1 5 6 4 0
  5 6 4 0 1 2 3
  4 0 6 2 3 1 5
  6 2 0 1 5 3 4
  1 5 3 4 0 6 2
  3 4 5 6 2 0 1
has 3 orthogonal mates
  0 1 2 3 4 5 6   0 1 2 3 4 5 6   0 1 2 3 4 5 6
  5 6 4 0 1 2 3   3 4 5 6 2 0 1   6 2 0 1 5 3 4
  1 5 3 4 0 6 2   4 0 6 2 3 1 5   3 4 5 6 2 0 1
  6 2 0 1 5 3 4   2 3 1 5 6 4 0   1 5 3 4 0 6 2
  3 4 5 6 2 0 1   5 6 4 0 1 2 3   2 3 1 5 6 4 0
  2 3 1 5 6 4 0   6 2 0 1 5 3 4   4 0 6 2 3 1 5
  4 0 6 2 3 1 5   1 5 3 4 0 6 2   5 6 4 0 1 2 3
so a(7)=3. (End)

Natalia Makarova, Diagonal Latin square with 10 orthogonal squares
Natalia Makarova, DB CF ODLS of order 9
Natalia Makarova, Maximum number of normalized ODLS from one DLS
Natalia Makarova, Comments for result a(12) >= 3855983322
Natalia Makarova, New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square
Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, square of order 9 with 516 orthogonal squares (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, About the diagonal Latin square of order 12 with 1764493860 orthogonal diagonal mates (in Russian).
Eduard I. Vatutin, Duplicate solutions removing using parallel and distributed DLX (in Russian).
Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition — 2021, pp. 77-79. (in Russian)
Eduard I. Vatutin, Proving list (best known examples).
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
Eduard I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk and V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina and Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
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.
Index entries for sequences related to Latin squares and rectangles.

Cf. A001438, A274171, A305571, A328873.

Definition corrected by Max Alekseyev, Dec 07 2019
a(9) added by Eduard I. Vatutin, Dec 12 2020
Edited by Max Alekseyev, Apr 01 2022

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

A107431 Triangle read by rows: T(n,k) = maximal number of rounds for the social golfer problem with n groups of k golfers (n >= 2, 2 <= k <= n).

Original entry on oeis.org

3, 5, 4, 7, 4, 5, 9, 7, 5, 6, 11, 8, 7, 6, 3, 13, 10, 9

Offset: 2

N. J. A. Sloane, following a tip from Ed Pegg Jr, May 28 2005

The problem is to find the largest number of rounds of golf that can be arranged with n*k golfers who play in n groups of k. No golfer may play in the same group as any other golfer twice (i.e., maximum socialization is achieved).
T(6,6) cannot be 4 since this would be equivalent to a pair of mutually orthogonal Latin squares of order 6.
T(n,k) = 1 for values of n and k outside this range.
The next term T(7,5) is known to be either 7 or 8.
T(n,n) = A001438(n) + 2. - Floris P. van Doorn, Sep 05 2019

			Triangle begins:
   3;
   5, 4;
   7, 4, 5;
   9, 7, 5, 6;
  11, 8, 7, 6, 3;
  ...
T(2,2) = 3 from { 12/34, 13/24, 14/23 }.

W. Harvey, Results Page for the Social Golfer Problem
Floris P. van Doorn, Dagstuhl's Happy Diner Problem - Latin Squares

Column 3 gives A107432.

Cf. A001438.

A091261 Number of orthogonal mates for the cyclic Latin squares of odd order.

Original entry on oeis.org

1, 3, 635, 2049219, 7372235460687

Offset: 3

Ian Wanless, Feb 23 2004

It is well known that the Cayley table of the cyclic group of order n has no orthogonal mate whenever n is even. This sequence reports the number of standardized orthogonal mates when n is odd, starting at n=3 (the case n=1 being meaningless). The word "standardized" refers to the fact that we only count mates which have their first row in natural order, since relabeling of the symbols in the orthogonal mate does not affect its defining property.

N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.
B. M. Maenhaut and I. M. Wanless, Atomic Latin squares of order eleven, J. Combin. Designs, Vol. 12 (2004), pp. 12-34.

Cf. A001438.

a(n) grows at least exponentially in n [Cavenagh, Wanless]. - Ian Wanless, Jul 30 2010

cp's OEIS Frontend

A287695 Maximum number of diagonal Latin squares with the first row in ascending order that can be orthogonal to a given diagonal Latin square 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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A107431 Triangle read by rows: T(n,k) = maximal number of rounds for the social golfer problem with n groups of k golfers (n >= 2, 2 <= k <= n).

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A091261 Number of orthogonal mates for the cyclic Latin squares of odd order.

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