A340545 -id:A340545 - 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!.

A293778 Number of centrally symmetric diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 960, 0, 14192640, 5449973760, 118753937326080

Offset: 1

Eduard I. Vatutin, Oct 16 2017

Centrally symmetric diagon 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] (numbering of rows and columns from 0 to n-1).
It seems that a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist) and 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 A292517(n) <= a(4n). - Eduard I. Vatutin, May 26 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

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, 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)
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. A292516, A292517, A293777, A340545.

a(n) = A293777(n) * n!.

A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 47, 0, 0, 0

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A doubly symmetric square has symmetries in both the horizontal and vertical planes (see A292517).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A340545(n). - Eduard I. Vatutin, May 28 2021

			An example of a doubly symmetric diagonal Latin square:
  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.

Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-11 (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.

Cf. A287764, A292517, A287650, A340545, A340546.

Name clarified by Andrew Howroyd, Oct 22 2023

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

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A293778 Number of centrally symmetric diagonal Latin squares of order n.

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A340550 Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

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