

A287649


Number of horizontally symmetric diagonal Latin squares of order 2n with constant first row.


4




OFFSET

1,2


COMMENTS

The number of horizontally symmetric diagonal Latin squares with constant first row (X) is equal to number of vertically symmetric diagonal Latin squares with constant first row. Total number of symmetric diagonal Latin squares with constant first row is equal to 2*XY, where Y is a number of doubly symmetric diagonal Latin squares with constant first row (sequence A287650).  Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Sum of symmetric elements a[i][j] and a[i][n1j] in horizontally symmetric normalized square is constant and equal to n1 for all pairs of elements (numbering of rows and columns from 0 to n1, values of elements of square also from 0 to n1). This is not true for vertically symmetric normalized square.  Eduard I. Vatutin, Oct 19 2017


LINKS

Table of n, a(n) for n=1..5.
S. E. Kochemazov, E. I. Vatutin, 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, S. E. Kochemazov, O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98100 (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, Discussion about properties of diagonal Latin squares at forum.boinc.ru, continuation (in Russian)
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) = A292516(n)/n!.


EXAMPLE

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


CROSSREFS

Cf. A003191, A287650, A292516.
Sequence in context: A139772 A092238 A228252 * A229352 A229815 A273498
Adjacent sequences: A287646 A287647 A287648 * A287650 A287651 A287652


KEYWORD

nonn,more,changed


AUTHOR

Eduard I. Vatutin, May 29 2017


EXTENSIONS

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


STATUS

approved



