A287649 Number of horizontally symmetric diagonal Latin squares of order 2n with the first row in ascending order.

0, 2, 64, 3612672, 82731715264512

Offset: 1

Author

Eduard I. Vatutin, May 29 2017

Keywords

• nonn
• more
• hard

Comments

The number of horizontally symmetric diagonal Latin squares (X) is equal to the number of vertically symmetric diagonal Latin squares. The total number of diagonal Latin squares with either horizontal or vertical symmetry (see A296060) is equal to 2*X-Y, where Y is the number of doubly symmetric diagonal Latin squares (see A287650). - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017

The sum of symmetric elements a[i, j] and a[i, n-1-j] in a horizontally symmetric normalized square of order n is constant and equal to n-1 for all pairs of elements (with rows and columns numbered from 0 to n-1 and elements values from 0 to n-1). This is not true for vertically symmetric normalized squares. - Eduard I. Vatutin, Oct 19 2017

Examples

			Horizontally symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 0 5 3 1
  5 4 3 2 1 0
  2 5 4 1 0 3
  3 0 1 4 5 2
  1 3 5 0 2 4
Vertically symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 5 0 3 1
  3 5 1 2 0 4
  5 3 0 4 1 2
  2 4 3 1 5 0
  1 0 4 5 2 3
		

Links

• S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.
• E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
• E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, continuation (in Russian)
• E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian)
• E. I. Vatutin, S. E. Kochemazov, and 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, and 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, and 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, and 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, A287650, A292516, A296060, A296061, A340546.

Formula

a(n) = A292516(n) / (2*n)!.

a(n) = (A296060(n) + A287650(n/2))/2 for even n; a(n) = A296060(n)/2 for odd n. - Andrew Howroyd, May 28 2021

Extensions

a(5) calculated and added by Eduard I. Vatutin, S. E. Kochemazov and O. S. Zaikin, Jun 15 2017