A292517 Number of doubly symmetric diagonal Latin squares of order 4n.

48, 495452160, 38903149816763645952000, 127654439655255918929515331054014121902080000

Offset: 1

Author

Eduard I. Vatutin, Sep 18 2017

Keywords

• nonn
• more
• hard

Comments

A doubly symmetric square has symmetries in both horizontal and vertical planes.

The plane symmetry requires one-to-one correspondence between the values of elements a[i,j] and a[N+1-i,j] in a vertical plane, and between the values of elements a[i,j] and a[i,N+1-j] in a horizontal plane for 1 <= i,j <= N. - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017

Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A293778(4n). - Eduard I. Vatutin, May 03 2021

Examples

			Doubly symmetric diagonal Latin square example:
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

• A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
• Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian).
• Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
• E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79.
• Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
• E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 17-19. (in Russian)
• E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Working on errors, Intellectual and Information Systems (2017), pp. 30-36. (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. A003191, A287649, A287650, A293778, A340550.

Formula

a(n) = A287650(n) * (4n)!.

Extensions

a(2) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017

Edited and a(3) from A287650 added by Max Alekseyev, Aug 23 2018, Sep 07 2018

a(4) from Andrew Howroyd, May 31 2021