A330391 -id:A330391 - cp's OEIS frontend

A305570 Number of diagonal Latin squares of order n with the first row in order and at least one orthogonal diagonal mate.

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 256, 632064, 95024976

Offset: 1

Eduard I. Vatutin, Jun 05 2018

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (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)
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.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and I. I. Citerra, Estimation of the probability of finding orthogonal diagonal Latin squares among general diagonal Latin squares, Recognition - 2018. Kursk: SWSU, 2018. pp. 72-74. (in Russian)
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).
Eduard I. Vatutin, Natalia N. Nikitina, and Maxim O. Manzuk, Additional calculated 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).
Eduard I. Vatutin, List of all main classes of orthogonal diagonal Latin squares of orders 1-8.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274171, A305568, A305571, A330391.

a(n) = A305571(n) / n!.

a(n) = A274171(n) - A305568(n).

Name clarified by Andrew Howroyd, Oct 19 2020

a(9) added by Eduard I. Vatutin, Dec 22 2020

A309210 Number of main classes of extended self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 5, 23, 18865, 33240

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.
A333366(n) <= A329685(n) <= a(n) <= A330391(n). - Eduard I. Vatutin, Jun 07 2020
a(10) >= 33240. - Eduard I. Vatutin, Jul 09 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 the 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.
E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of order 9.
E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of order 10.
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. 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).
Eduard Vatutin and Oleg Zaikin, Classification of Cells Mapping Schemes Related to Orthogonal Diagonal Latin Squares of Small Order, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 21-34.
Index entries for sequences related to Latin squares and rectangles

Cf. A287761, A309598, A309599, A329685, A333366, A330391.

a(9) added by Eduard I. Vatutin, Dec 08 2020

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

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

A333366 Number of main classes of doubly self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 8, 88, 0

Offset: 1

Eduard I. Vatutin, Mar 17 2020

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

a(n) <= A329685(n) <= A309210(n) <= A330391(n). - Eduard I. Vatutin, Jun 01 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, 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. A329685, A309210, A333367.

a(10) = 0 is established by Lu et al. (2011).

A337309 Number of main classes of bachelor diagonal Latin squares of order n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 967, 4871991, 3292326080087

Offset: 1

Eduard I. Vatutin, Aug 22 2020

A bachelor diagonal Latin square is one with no orthogonal mate.

Eduard I. Vatutin, About the number of main classes of bachelor diagonal Latin squares of orders 1-8 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, 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).
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. A305568, A305569, A330391, A287764.

a(n) = A287764(n) - A330391(n).

a(9) added by Eduard I. Vatutin, Dec 08 2020

cp's OEIS Frontend

A305570 Number of diagonal Latin squares of order n with the first row in order and at least one orthogonal diagonal mate.

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A309210 Number of main classes of extended self-orthogonal diagonal Latin squares of order n.

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

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A333366 Number of main classes of doubly self-orthogonal diagonal Latin squares of order n.

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A337309 Number of main classes of bachelor diagonal Latin squares of order n.

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