

A287650


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


6




OFFSET

1,1


COMMENTS

A doubly symmetric square has symmetries in both the horizontal and vertical planes.
The plane symmetry requires onetoone correspondence between the values of elements a[i,j] and a[N+1i,j] in a vertical plane, and between the values of elements a[i,j] and a[i,N+1j] 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*n3))) 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


LINKS

Table of n, a(n) for n=1..4.
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. 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.
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, 38.
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).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
Index entries for sequences related to Latin squares and rectangles


FORMULA

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


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
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)


CROSSREFS

Cf. A001147, A003191, A287649, A292517, A293777, A340550.
Sequence in context: A128122 A082178 A345396 * A082912 A265013 A083973
Adjacent sequences: A287647 A287648 A287649 * A287651 A287652 A287653


KEYWORD

nonn,more,hard


AUTHOR

Eduard I. Vatutin, May 29 2017


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


STATUS

approved



