A309210 -id:A309210 - cp's OEIS frontend

A309598 Number of extended self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 4, 0, 256, 4608, 24437088, 510566400

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.
A333367(n) <= A287761(n) <= a(n) <= A305570(n). - Eduard I. Vatutin, Jun 07 2020
a(10) >= 510566400. - 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, A305570, A309210, A309599, A333367.

From Eduard I. Vatutin, Feb 25 2020: (Start)
a(n) = A287761(n) for 1 <= n <= 6.
a(n) = 4*A287761(n) for 7 <= n <= 8. (End)
a(10) = A309210(10)*A299784(10) because no DSODLS exist for order n=10 and no ESODLS of order n=10 have generalized M-symmetries (automorphisms). - Eduard I. Vatutin, Jul 10 2020

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

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

Original entry on oeis.org

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

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).

A383684 Minimum number of transversals in an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 0, 23, 128, 133, 716

Offset: 1

Eduard I. Vatutin, May 05 2025

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.

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A090741, A091323, A287644, A287645, A309210, A309598, A309599, A357514, A387124, A387187.

A387124 Maximum number of transversals in an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 0, 133, 384, 2241, 988

Offset: 1

Eduard I. Vatutin, Aug 17 2025

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.

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A090741, A091323, A287644, A287645, A309210, A309598, A309599, A357514, A383684, A387187.

A382952 Maximum number of intercalates in an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 0, 0, 18, 112, 72, 53

Offset: 1

Eduard I. Vatutin, Apr 09 2025

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.
Table showing minimums and maximums:
  order                      |  4  5  6  7   8   9  10
  min number of intercalates | 12  0  -  0  16   0   5
  max number of intercalates | 12  0  - 18 112  72  53 (this sequence)

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving list, minimum number of intercalates (best known examples).
Eduard I. Vatutin, Proving list, maximum number of intercalates (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307164, A309210, A309598, A309599, A360223, A382957.

A387187 a(n) is the number of distinct numbers of transversals an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 5, 244, 62

Offset: 1

Eduard I. Vatutin, Aug 21 2025

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.

			For n=8 the number of transversals that an extended self-orthogonal diagonal Latin square of order 7 may have is 128, 192, 224, 256, or 384. Since there are 3 distinct values, a(8)=5.

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving lists (1, 4, 5, 7, 8, 9, 10).
Eduard I. Vatutin, Graphical representation of the spectra.
Index entries for sequences related to Latin squares and rectangles.

Cf. A309210, A309598, A309599, A344105, A350585, A383684, A387124.

A387236 Minimum number of diagonal transversals in an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 0, 8, 16, 15, 75

Offset: 1

Eduard I. Vatutin, Aug 23 2025

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.

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A309210, A309598, A309599, A354068, A360220.

A382957 a(n) is the number of distinct numbers of intercalates in an extended self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 3, 8, 52, 45

Offset: 1

Eduard I. Vatutin, Apr 10 2025

A self-orthogonal diagonal Latin square (SODLS, see A329685) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS, see A309210) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

			For n=7 the number of intercalates that an extended self-orthogonal diagonal Latin square of order 7 may have is 0, 10, or 18. Since there are 3 distinct values, a(7)=3.

Eduard I. Vatutin, Proving lists (1, 4, 5, 7, 8, 9, 10).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A309210, A309598, A309599, A329685, A345760, A382952.

cp's OEIS Frontend

A309598 Number of extended self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

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A309599 Number 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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A383684 Minimum number of transversals in an extended self-orthogonal diagonal Latin square of order n.

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A387124 Maximum number of transversals in an extended self-orthogonal diagonal Latin square of order n.

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A382952 Maximum number of intercalates in an extended self-orthogonal diagonal Latin square of order n.

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A387187 a(n) is the number of distinct numbers of transversals an extended self-orthogonal diagonal Latin square of order n.

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A387236 Minimum number of diagonal transversals in an extended self-orthogonal diagonal Latin square of order n.

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A382957 a(n) is the number of distinct numbers of intercalates in an extended self-orthogonal diagonal Latin squares of order n.

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