A287762 -id:A287762 - cp's OEIS frontend

A309599 Number of extended self-orthogonal diagonal Latin squares of order n.

1, 0, 0, 48, 480, 0, 1290240, 185794560, 8867730493440, 1852743352320000

Offset: 1

Eduard I. Vatutin, Aug 09 2019

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.
A333671(n) <= A287762(n) <= a(n) <= A305571(n). - Eduard I. Vatutin, Jun 07 2020
a(10) >= 1852743352320000. - Eduard I. Vatutin, Jul 10 2020

			The diagonal Latin square
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 5 7 9 8 6 3
  5 0 1 6 3 9 8 2 4 7
  9 3 5 8 2 1 7 4 0 6
  4 6 3 5 7 8 0 9 2 1
  8 4 6 9 1 3 2 5 7 0
  7 8 9 0 6 4 5 1 3 2
  2 9 4 7 8 0 3 6 1 5
  6 5 7 1 0 2 4 3 9 8
  3 7 8 2 9 6 1 0 5 4
has orthogonal diagonal Latin square
  0 1 2 3 4 5 6 7 8 9
  3 5 9 8 6 2 0 1 4 7
  4 3 8 7 2 1 9 0 5 6
  6 9 3 4 8 0 1 2 7 5
  7 2 0 1 9 3 5 8 6 4
  2 0 1 5 7 6 4 9 3 8
  8 6 4 2 0 9 7 5 1 3
  1 7 6 0 5 4 8 3 9 2
  9 8 5 6 1 7 3 4 2 0
  5 4 7 9 3 8 2 6 0 1
from the same main class.

E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian).
E. I. Vatutin, About the lower bound of number of ESODLS of order 10 (in Russian).
E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of orders 1-8.
Eduard 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 and A. D. Belyshev, About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (in Russian)
E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
Index entries for sequences related to Latin squares and rectangles

Cf. A287761, A287762, A309210, A309598, A305571, A333671.

a(9) calculated by Eduard I. Vatutin, Dec 08 2020, independently checked by Oleg S. Zaikin, Dec 16 2024, added by Eduard I. Vatutin, Jan 30 2025

a(10) added by Eduard I. Vatutin, Oleg S. Zaikin, Jan 30 2025

A287761 Number of self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 64, 1152, 224832, 234255360

Offset: 1

Eduard I. Vatutin, May 31 2017

A self-orthogonal diagonal Latin square is a diagonal Latin square orthogonal to its transpose.

A333367(n) <= a(n) <= A309598(n) <= A305570(n). - Eduard I. Vatutin, Apr 26 2020

			0 1 2 3 4 5 6 7 8 9
5 2 0 9 7 8 1 4 6 3
9 5 7 1 8 6 4 3 0 2
7 8 6 4 9 2 5 1 3 0
8 9 5 0 3 4 2 6 7 1
3 6 9 5 2 1 7 0 4 8
4 3 1 7 6 0 8 2 9 5
6 7 8 2 5 3 0 9 1 4
2 0 4 6 1 9 3 8 5 7
1 4 3 8 0 7 9 5 2 6

E. I. Vatutin, About the number of SODLS of order 10, a(10) value is wrong (in Russian).
E. I. Vatutin, About the number of SODLS of order 10, corrected value a(10) (in Russian).
E. I. Vatutin, List of all main classes of self-orthogonal diagonal Latin squares of orders 1-10.
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 and A. D. Belyshev, About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (in Russian)
E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
Harry White, Self-orthogonal Diagonal Latin Squares. How many.
Index entries for sequences related to Latin squares and rectangles.

Cf. A160368, A287762, A329685.

a(n) = A287762(n)/n!.
From Eduard I. Vatutin, Mar 14 2020: (Start)
a(i) != A329685(i)*A299784(i)/2 for i=1..9 due to the existence of doubly self-orthogonal diagonal Latin square (DSODLS) and/or generalized symmetries (automorphisms) for some SODLS.
a(10) = A329685(10)*A299784(10)/2 because no DSODLS exist for order n=10 and no SODLS of order n=10 have generalized symmetries (automorphisms). (End)

a(10) from Eduard I. Vatutin, Mar 14 2020

a(10) corrected by Eduard I. Vatutin, Apr 24 2020

A329685 Number of main classes of self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 8, 470, 30502

Offset: 1

Eduard I. Vatutin, Feb 25 2020

A self-orthogonal diagonal Latin square is a diagonal Latin square orthogonal to its transpose.

A333366(n) <= a(n) <= A309210(n) <= A330391(n). - Eduard I. Vatutin, Apr 26 2020

			0 1 2 3 4 5 6 7 8 9
5 2 0 9 7 8 1 4 6 3
9 5 7 1 8 6 4 3 0 2
7 8 6 4 9 2 5 1 3 0
8 9 5 0 3 4 2 6 7 1
3 6 9 5 2 1 7 0 4 8
4 3 1 7 6 0 8 2 9 5
6 7 8 2 5 3 0 9 1 4
2 0 4 6 1 9 3 8 5 7
1 4 3 8 0 7 9 5 2 6

A. D. Belyshev, List of 30502 essentially distinct self-orthogonal diagonal Latin squares of order 10
E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian).
E. I. Vatutin, About the number of main classes for SODLS of order 9 (in Russian).
E. I. Vatutin, About the number of SODLS of order 10 (in Russian).
E. I. Vatutin, List of all main classes of self-orthogonal diagonal Latin squares of orders 1-10.
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 and A. D. Belyshev, About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (in Russian)
E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
Eduard I. Vatutin, Natalia N. Nikitina, and Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Index entries for sequences related to Latin squares and rectangles

Cf. A309210, A287761, A287762.

a(9) from Eduard I. Vatutin, Mar 12 2020
a(10) from Eduard I. Vatutin, Mar 14 2020
a(10) corrected by Natalia Makarova, Apr 10 2020

A333671 Number of doubly self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 480, 0, 322560, 46448640, 10381271040, 0

Offset: 1

Eduard I. Vatutin, Apr 01 2020

A doubly self-orthogonal diagonal Latin square (DSODLS) is a diagonal Latin square orthogonal to its transpose and antitranspose.

a(n) <= A287762(n) <= A309599(n) <= A305571(n). - Eduard I. Vatutin, Jun 06 2020

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

R. Lu, S. Liu, and J. Zhang, Searching for Doubly Self-orthogonal Latin Squares. Lecture Notes in Computer Science 6876 (2011), 538-545.
E. I. Vatutin, About the number of DSODLS of orders 1-10 (in Russian).
E. I. Vatutin, List of all main classes of doubly self-orthogonal diagonal Latin squares of orders 1-10.
E. I. Vatutin and A. D. Belyshev, About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (in Russian)
E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
Index entries for sequences related to Latin squares and rectangles.

Cf. A287762, A309599.

cp's OEIS Frontend

A309599 Number of extended self-orthogonal diagonal Latin squares of order n.

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A287761 Number of self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

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A329685 Number of main classes of self-orthogonal diagonal Latin squares of order n.

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A333671 Number of doubly self-orthogonal diagonal Latin squares of order n.

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