A287650 Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.

2, 12288, 81217160478720, 6101215007109090122576072540160

Offset: 1

Author

Eduard I. Vatutin, May 29 2017

Keywords

• nonn
• more
• hard

Comments

A doubly symmetric square has symmetries in both the 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) <= A293777(4n). - Eduard I. Vatutin, May 26 2021

a(n)/(A001147(n)*2^(n*(4*n-3))) is the number of 2n X 2n grids with two instances of each of 1..n on the main diagonal and in each row and column with the first row in nondescreasing order. - Andrew Howroyd, May 30 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
Reflection of all rows is equivalent to the exchange of elements 0 and 7, 1 and 6, 2 and 5, 3 and 4; hence, this square is horizontally symmetric. Reflection of all columns is equivalent to the exchange of elements 0 and 1, 2 and 4, 3 and 5, 6 and 7; hence, the square is also vertically symmetric.
From _Andrew Howroyd_, May 30 2021: (Start)
a(2) = 4*3*1024 = 12288. The 4 base quarter square arrangements are:
  1 1 2 2  1 1 2 2  1 1 2 2  1 1 2 2
  1 2 1 2  1 2 2 1  2 2 1 1  2 2 1 1
  2 1 2 1  2 2 1 1  1 1 2 2  2 2 1 1
  2 2 1 1  2 1 1 2  2 2 1 1  1 1 2 2
(End)
		

Links

• A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
• E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, value a(4) is wrong (in Russian)
• E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian)
• E. I. Vatutin, S. E. Kochemazov, 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, 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).
• 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. A001147, A003191, A287649, A292517, A293777, A340550.

Formula

a(n) = A292517(n) / (4n)!.

Extensions

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

Edited and a(3) from Alexey D. Belyshev added by Max Alekseyev, Aug 23 2018, Sep 07 2018

a(4) from Andrew Howroyd, May 31 2021