A292517 -id:A292517 - cp's OEIS frontend

A287650 Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.

2, 12288, 81217160478720, 6101215007109090122576072540160

Offset: 1

Eduard I. Vatutin, May 29 2017

A doubly symmetric square has symmetries in both the 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) <= A293777(4n). - Eduard I. Vatutin, May 26 2021
a(n)/(A001147(n)*2^(n*(4*n-3))) 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

			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)

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. 98-100 (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).
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. A001147, A003191, A287649, A292517, A293777, A340550.

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

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

A292516 Number of horizontally symmetric diagonal Latin squares of order 2n.

Original entry on oeis.org

0, 48, 46080, 145662935040, 300216848351861145600

Offset: 1

Eduard I. Vatutin, Sep 18 2017

The number of horizontally symmetric diagonal Latin squares (X) is equal to the number of vertically symmetric diagonal Latin squares. The total number of symmetric diagonal Latin squares is equal to 2*X-Y, where Y is a number of double symmetric diagonal Latin squares (sequence A292517). - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017

			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

E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, On Some Features of Symmetric Diagonal Latin Squares, CEUR WS, vol. 1940 (2017), pp. 74-79.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and 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, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and 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, and 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, A287650, A287649, A296060, A296061, A340546.

a(n) = A287649(n) * (2*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!.

A296061 Number of one-plane symmetric diagonal Latin squares of order 2n.

Original entry on oeis.org

0, 96, 92160, 290830417920

Offset: 1

Eduard I. Vatutin, Dec 04 2017

One-plane symmetric diagonal Latin squares are vertically or horizontally symmetric diagonal Latin squares. a(n) is equal to 2*X-Y, where X is the number of horizontally symmetric diagonal Latin squares (sequence A292516), and Y is the number of doubly symmetric diagonal Latin squares (sequence A292517).

			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
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

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, Investigation of the properties of symmetric diagonal Latin squares. Corrections. Intellectual and Information Systems (2017), pp. 30-36 (in Russian)
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, 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, A292516, A292517, A296060, A340546.

a(n) = 2*A292516(n) - A292517(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

cp's OEIS Frontend

A287650 Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.

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A292516 Number of horizontally symmetric diagonal Latin squares of order 2n.

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

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A296061 Number of one-plane symmetric diagonal Latin squares of order 2n.

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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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