

A292517


Number of doubly symmetric diagonal Latin squares of order 2n.


4




OFFSET

1,2


COMMENTS

Doubly symmetric square has symmetries both in horizontal and vertical planes.
One plane symmetry requires onetoone correspondens between values of elements a[i][j] and a[Ni][j] in vertical plane and between values of elements a[i][j] and a[i][Nj] in horizontal plane for all i and j values (numbering if indexes from 1).  Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
It seems that doubly symmetric diagonal Latin squares exists only for orders N == 0 (mod 4).  Eduard I. Vatutin, Oct 18 2017


LINKS

Table of n, a(n) for n=1..4.
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, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 7479.
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. 1719 (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. 3036 (in Russian)
Index entries for sequences related to Latin squares and rectangles


FORMULA

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


EXAMPLE

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 onetoone 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.


CROSSREFS

Cf. A003191, A287649, A287650.
Sequence in context: A002834 A057380 A234565 * A036210 A257875 A037941
Adjacent sequences: A292514 A292515 A292516 * A292518 A292519 A292520


KEYWORD

nonn,more,bref


AUTHOR

Eduard I. Vatutin, Sep 18 2017


EXTENSIONS

a(4) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017


STATUS

approved



