A305570 -id:A305570 - 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

A305571 Number of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

Original entry on oeis.org

1, 0, 0, 48, 480, 0, 1290240, 25484820480, 34482663290880

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).
E. 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. A274806, A305569, A305570.

a(n) = A305570(n) * n!.

a(n) = A274806(n) - A305569(n).

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

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

A330391 Number of main classes of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 5, 1105, 75307

Offset: 1

Eduard I. Vatutin, Feb 25 2020

Natalia Makarova, Database CF ODLS of order n
E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian)
E. I. Vatutin, List of all main classes of orthogonal diagonal Latin squares of orders 1-8.
E. I. Vatutin, List of all main classes of orthogonal diagonal Latin squares of order 9.
E. I. Vatutin, List of known main classes of orthogonal diagonal Latin squares of order 11.
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, 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. A287651, A287764, A287695, A305570, A305571, A337309.

a(n) = A287764(n) - A337309(n).

a(9) added by Eduard I. Vatutin, Dec 12 2020

A349199 a(n) is the number of distinct numbers of diagonal transversals that an orthogonal diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 3, 31, 165

Offset: 1

Eduard I. Vatutin, Nov 10 2021

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A345370(n).

a(10) >= 390, a(11) >= 560, a(12) >= 13429. - Eduard I. Vatutin, Nov 10 2021, updated Jan 29 2023

			For n=8 the number of diagonal transversals that an orthogonal diagonal Latin square of order 8 may have is 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 32, 36, 38, 40, 42, 44, 48, 52, 56, 64, 72, 88, 96, or 120. Since there are 31 distinct values, a(8)=31.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 4, 5, 7, 8, 9, 10, 11, 12).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A305570, A345370.

A350585 a(n) is the number of distinct numbers of transversals that an orthogonal diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 25, 295

Offset: 1

Eduard I. Vatutin, Mar 27 2022

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A344105(n).

a(10) >= 193, a(11) >= 3588, a(12) >= 10465. - updated by Eduard I. Vatutin, Jan 29 2023

			For n=8 the number of transversals that an orthogonal diagonal Latin square of order 8 may have is 16, 32, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 88, 96, 112, 128, 132, 144, 160, 168, 192, 224, 256, 320, or 384. Since there are 25 distinct values, a(8)=25.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Examples (1, 4, 5, 7, 8, 9, 10, 11, 12).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A305570, A344105, A349199.

A360220 Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 0, 27, 120, 333

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An orthogonal diagonal Latin square is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) >= 866, a(11) >= 4828, a(12) >= 30192, a(13) >= 131106, a(17) >= 204995269, a(19) >= 11254190082.
For most orders n, at least one diagonal Latin square with the maximal number of diagonal transversals has an orthogonal mate and A287648(n) = a(n). Known exceptions: n=6 and n=10. - Eduard I. Vatutin, Feb 17 2023
Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= A354068(n) <= a(n) <= A287648(n). - Eduard I. Vatutin, Mar 04 2023

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A305570, A305571, A349199, A354068.

A305568 Number of bachelor diagonal Latin squares of order n with the first row in ascending order.

Original entry on oeis.org

0, 0, 0, 0, 4, 128, 170944, 7446955776, 5056994558482608

Offset: 1

Eduard I. Vatutin, Jun 05 2018

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

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. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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, 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, 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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274171, A266177, A305569, A305570.

a(n) = A305569(n) / n!.

a(n) = A274171(n) - A305570(n).

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

A360223 Maximum number of intercalates in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

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

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An intercalate is a 2 X 2 subsquare of a Latin square.
An orthogonal diagonal Latin square is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) >= 76, a(11) >= 94, a(12) >= 324, a(13) >= 26. - Eduard I. Vatutin, updated Feb 25 2024

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A305570, A305571, A307164, A354050, A360221.

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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A305571 Number of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

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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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A330391 Number of main classes of diagonal Latin squares of order n with at least one orthogonal diagonal mate.

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A349199 a(n) is the number of distinct numbers of diagonal transversals that an orthogonal diagonal Latin square of order n can have.

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A350585 a(n) is the number of distinct numbers of transversals that an orthogonal diagonal Latin square of order n can have.

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A360220 Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

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A305568 Number of bachelor 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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A360223 Maximum number of intercalates in an orthogonal diagonal Latin square of order n.

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