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

Author

Eduard I. Vatutin, May 30 2017

Keywords

• nonn
• more
• hard

Comments

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

Examples

			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)
		

Links

• 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.

Crossrefs

Cf. A001438, A274171, A305571, A328873.

Extensions

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