A293778 -id:A293778 - cp's OEIS frontend

A293777 Number of centrally symmetric diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 8, 0, 2816, 135168, 327254016

Offset: 1

Eduard I. Vatutin, Oct 16 2017

A centrally symmetric diagonal Latin square is a square with one-to-one correspondence between elements within all pairs a[i][j] and a[n-1-i][n-1-j] (with rows and columns numbered from 0 to n-1).
a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist). It seems that a(n)=0 for n == 2 (mod 4).
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(4n) >= A287650(n). - Eduard I. Vatutin, May 03 2021

			0 1 2 3 4 5 6 7 8
6 3 0 2 7 8 1 4 5
3 2 1 8 6 7 0 5 4
7 8 6 5 1 3 4 0 2
8 6 4 7 2 0 5 3 1
2 7 5 6 8 4 3 1 0
5 4 7 0 3 1 8 2 6
4 5 8 1 0 2 7 6 3
1 0 3 4 5 6 2 8 7

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Properties of central symmetry for diagonal Latin squares, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
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

Cf. A287649, A287650, A293778, A340545.

a(n) = A293778(n) / n!.

A292517 Number of doubly symmetric diagonal Latin squares of order 4n.

Original entry on oeis.org

48, 495452160, 38903149816763645952000, 127654439655255918929515331054014121902080000

Offset: 1

Eduard I. Vatutin, Sep 18 2017

A doubly symmetric square has symmetries in both 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) <= A293778(4n). - Eduard I. Vatutin, May 03 2021

			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 one-to-one 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.

A. D. Belyshev, Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4, 2017 (in Russian)
Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4) (in Russian).
Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (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)
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

Cf. A003191, A287649, A287650, A293778, A340550.

a(n) = A287650(n) * (4n)!.

a(2) corrected by Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Edited and a(3) from A287650 added by Max Alekseyev, Aug 23 2018, Sep 07 2018
a(4) from Andrew Howroyd, May 31 2021

A340546 Number of main classes of diagonal Latin squares of order 2n that contain a one-plane symmetric square.

Original entry on oeis.org

0, 1, 2, 9717

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A one-plane symmetric diagonal Latin square is a vertically or horizontally symmetric diagonal Latin square (see A296060). Such diagonal Latin squares do not exist for odd orders > 1.

			A 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
A 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
Both are one-plane symmetric diagonal Latin squares.

Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-8 (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.

Cf. A287649, A287764, A292516, A293777, A293778, A296060, A296061, A340550.

Name clarified by Andrew Howroyd, Oct 22 2023

A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 32, 301, 430090

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A centrally symmetric diagonal Latin square is a square with one-to-one correspondence between elements within all pairs a[i, j] and a[n-1-i, n-1-j] (with numbering of rows and columns from 0 to n-1).
It seems that a(n)=0 for n==2 (mod 4).
Centrally symmetric Latin squares are Latin squares, so a(n) <= A287764(n).
The canonical form (CF) of a square is the lexicographically minimal item within the corresponding main class of diagonal Latin square.
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A340550(n) <= a(n). - Eduard I. Vatutin, May 28 2021

			For n=4 there is a single CF:
  0 1 2 3
  2 3 0 1
  3 2 1 0
  1 0 3 2
so a(4)=1.
For n=5 there are two different CFs:
  0 1 2 3 4   0 1 2 3 4
  2 3 4 0 1   1 3 4 2 0
  4 0 1 2 3   4 2 1 0 3
  1 2 3 4 0   2 0 3 4 1
  3 4 0 1 2   3 4 0 1 2
so a(5)=2.
Example of a centrally symmetric diagonal Latin square of order n=9:
  0 1 2 3 4 5 6 7 8
  6 3 0 2 7 8 1 4 5
  3 2 1 8 6 7 0 5 4
  7 8 6 5 1 3 4 0 2
  8 6 4 7 2 0 5 3 1
  2 7 5 6 8 4 3 1 0
  5 4 7 0 3 1 8 2 6
  4 5 8 1 0 2 7 6 3
  1 0 3 4 5 6 2 8 7

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, Properties of central symmetry for diagonal Latin squares, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
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)
E. I. Vatutin, About the number of main classes of centrally symmetric diagonal Latin squares of orders 1-9 (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.

Cf. A293777, A293778, A287764, A340550.

cp's OEIS Frontend

A293777 Number of centrally symmetric diagonal Latin squares of order n with the first row in ascending order.

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A292517 Number of doubly symmetric diagonal Latin squares of order 4n.

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A340546 Number of main classes of diagonal Latin squares of order 2n that contain a one-plane symmetric square.

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A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

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