A309598 Number of extended self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 4, 0, 256, 4608, 24437088, 510566400

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.

A333367(n) <= A287761(n) <= a(n) <= A305570(n). - Eduard I. Vatutin, Jun 07 2020

a(10) >= 510566400. - Eduard I. Vatutin, Jul 10 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 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.
• 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. I. Vatutin and A. D. Belyshev, About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (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.
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

Cf. A287761, A305570, A309210, A309599, A333367.

Formula

From Eduard I. Vatutin, Feb 25 2020: (Start)

a(n) = A287761(n) for 1 <= n <= 6.

a(n) = 4*A287761(n) for 7 <= n <= 8. (End)

a(10) = A309210(10)*A299784(10) because no DSODLS exist for order n=10 and no ESODLS of order n=10 have generalized M-symmetries (automorphisms). - Eduard I. Vatutin, Jul 10 2020

Extensions

a(9) calculated by Eduard I. Vatutin, Dec 08 2020, independently checked by Oleg S. Zaikin, Dec 16 2024, added by Eduard I. Vatutin, Jan 30 2025

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