A309598 -id:A309598 - cp's OEIS frontend

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

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

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

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

A333367 Number of doubly 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, 28608, 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) <= A287761(n) <= A309598(n) <= A305570(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, 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. A333366, A287761.

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

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

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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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A333367 Number of doubly self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

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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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