A309210 Number of main classes of extended self-orthogonal diagonal Latin squares of order n.

1, 0, 0, 1, 1, 0, 5, 23, 18865, 33240

Offset: 1

Author

Eduard I. Vatutin, Aug 09 2019

Keywords

• nonn
• more
• hard

Comments

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

A333366(n) <= A329685(n) <= a(n) <= A330391(n). - Eduard I. Vatutin, Jun 07 2020

a(10) >= 33240. - Eduard I. Vatutin, Jul 09 2020

Examples

			The diagonal Latin square
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 5 7 9 8 6 3
  5 0 1 6 3 9 8 2 4 7
  9 3 5 8 2 1 7 4 0 6
  4 6 3 5 7 8 0 9 2 1
  8 4 6 9 1 3 2 5 7 0
  7 8 9 0 6 4 5 1 3 2
  2 9 4 7 8 0 3 6 1 5
  6 5 7 1 0 2 4 3 9 8
  3 7 8 2 9 6 1 0 5 4
has the orthogonal diagonal Latin square
  0 1 2 3 4 5 6 7 8 9
  3 5 9 8 6 2 0 1 4 7
  4 3 8 7 2 1 9 0 5 6
  6 9 3 4 8 0 1 2 7 5
  7 2 0 1 9 3 5 8 6 4
  2 0 1 5 7 6 4 9 3 8
  8 6 4 2 0 9 7 5 1 3
  1 7 6 0 5 4 8 3 9 2
  9 8 5 6 1 7 3 4 2 0
  5 4 7 9 3 8 2 6 0 1
from the same main class.
		

Links

• E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian)
• E. I. Vatutin, About the lower bound of number of ESODLS of order 10 (in RUssian).
• E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of orders 1-8.
• E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of order 9.
• E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of order 10.
• Eduard I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
• E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
• 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).
• Eduard Vatutin and Oleg Zaikin, Classification of Cells Mapping Schemes Related to Orthogonal Diagonal Latin Squares of Small Order, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 21-34.
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

Cf. A287761, A309598, A309599, A329685, A333366, A330391.

Extensions

a(9) added by Eduard I. Vatutin, Dec 08 2020

a(10) added by Eduard I. Vatutin, Oleg S. Zaikin, Jan 30 2025