A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

1, 0, 0, 1, 0, 0, 0, 47, 0, 0, 0

Offset: 1

Author

Eduard I. Vatutin, Jan 11 2021

Keywords

• nonn
• more
• hard

Comments

A doubly symmetric square has symmetries in both the horizontal and vertical planes (see A292517).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A340545(n). - Eduard I. Vatutin, May 28 2021

Examples

			An example of a doubly symmetric diagonal Latin square:
  0 1 2 3 4 5 6 7
  3 2 7 6 1 0 5 4
  2 3 1 0 7 6 4 5
  6 7 5 4 3 2 0 1
  7 6 3 2 5 4 1 0
  4 5 0 1 6 7 2 3
  5 4 6 7 0 1 3 2
  1 0 4 5 2 3 7 6
In the horizontal direction there is a one-to-one correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4.
In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5.
		

Links

• Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-11 (in Russian).
• Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
• E. 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)
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A287764, A292517, A287650, A340545, A340546.

Extensions

Name clarified by Andrew Howroyd, Oct 22 2023