A305571 -id:A305571 - 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

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

A287695 Maximum number of diagonal Latin squares with the first row in ascending order that can be orthogonal to a given diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 3, 824, 614

Offset: 1

Eduard I. Vatutin, May 30 2017

A Latin square is normalized if in the first row elements come in increasing order. Any diagonal Latin square orthogonal to a given one can be normalized by renaming its elements (which does not break diagonality and orthogonality). - Max Alekseyev, Dec 07 2019
For all orders n>3 there are diagonal Latin squares without orthogonal mates (also known as bachelor squares), so the minimum number of diagonal Latin squares that can be orthogonal to the same diagonal Latin square is zero. For order n=1 the single square is orthogonal to itself. For n=2 and n=3 diagonal Latin squares do not exist (see A274171). For n=6 orthogonal diagonal Latin squares do not exist (see A305571), so a(6)=0. - Eduard I. Vatutin, May 03 2021
a(n) >= A328873(n) - 1. - Eduard I. Vatutin, Mar 29 2021
a(10) >= 10 (Updated). - Eduard I. Vatutin, Apr 27 2018
a(11) >= 32462. - Eduard I. Vatutin from T. Brada, Mar 11 2021
a(12) >= 3855983322. The result belongs to DLS, which has 30192 diagonal transversals. Calculations performed by a volunteer. - Natalia Makarova, Tomáš Brada, Nov 11 2021
a(13) >= 248703. - Natalia Makarova, Tomáš Brada, Apr 29 2021
a(14) >= 307662. - Natalia Makarova, Alex Chernov, Harry White, May 21 2021
a(16) >= 1658880, a(17) >= 2453352, a(18) >= 96, a(19) >= 1383, a(20) >= 995328, a(21) >= 995328, a(22) >= 432000, a(23) >= 525, a(24) >= 345600, a(25) >= 345600, a(26) >= 48, a(27) >= 345600, a(28) >= 663552, a(29) >= 663552, a(30) >= 40320. For values up to a(100), see the specified link "New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square". - Natalia Makarova, Alex Chernov, Harry White, Dec 06 2021

			From _Eduard I. Vatutin_, Mar 29 2021: (Start)
One of the best existing diagonal Latin squares of order 7
  0 1 2 3 4 5 6
  2 3 1 5 6 4 0
  5 6 4 0 1 2 3
  4 0 6 2 3 1 5
  6 2 0 1 5 3 4
  1 5 3 4 0 6 2
  3 4 5 6 2 0 1
has 3 orthogonal mates
  0 1 2 3 4 5 6   0 1 2 3 4 5 6   0 1 2 3 4 5 6
  5 6 4 0 1 2 3   3 4 5 6 2 0 1   6 2 0 1 5 3 4
  1 5 3 4 0 6 2   4 0 6 2 3 1 5   3 4 5 6 2 0 1
  6 2 0 1 5 3 4   2 3 1 5 6 4 0   1 5 3 4 0 6 2
  3 4 5 6 2 0 1   5 6 4 0 1 2 3   2 3 1 5 6 4 0
  2 3 1 5 6 4 0   6 2 0 1 5 3 4   4 0 6 2 3 1 5
  4 0 6 2 3 1 5   1 5 3 4 0 6 2   5 6 4 0 1 2 3
so a(7)=3. (End)

Natalia Makarova, Diagonal Latin square with 10 orthogonal squares
Natalia Makarova, DB CF ODLS of order 9
Natalia Makarova, Maximum number of normalized ODLS from one DLS
Natalia Makarova, Comments for result a(12) >= 3855983322
Natalia Makarova, New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square
Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, square of order 9 with 516 orthogonal squares (in Russian).
Eduard I. Vatutin, About the A328873(N)-1 <= A287695(N) inequality between the maximum cardinality of clique and the maximum number of orthogonal normalized mates for one diagonal Latin square (in Russian).
Eduard I. Vatutin, About the diagonal Latin square of order 12 with 1764493860 orthogonal diagonal mates (in Russian).
Eduard I. Vatutin, Duplicate solutions removing using parallel and distributed DLX (in Russian).
Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition — 2021, pp. 77-79. (in Russian)
Eduard I. Vatutin, Proving list (best known examples).
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
Eduard I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk and V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (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).
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)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
Index entries for sequences related to Latin squares and rectangles.

Cf. A001438, A274171, A305571, A328873.

Definition corrected by Max Alekseyev, Dec 07 2019
a(9) added by Eduard I. Vatutin, Dec 12 2020
Edited by Max Alekseyev, Apr 01 2022

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

A287762 Number of self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 480, 0, 322560, 46448640, 81587036160, 850065850368000

Offset: 1

Eduard I. Vatutin, May 31 2017

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

A333671(n) <= a(n) <= A309599(n) <= A305571(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.
H. White, Self-orthogonal Diagonal Latin Squares. How many.
Index entries for sequences related to Latin squares and rectangles.

Cf. A160368, A287761, A329685.

a(n) = A287761(n)*n!.

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

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

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.

A305569 Number of bachelor diagonal Latin squares of order n.

Original entry on oeis.org

0, 0, 0, 0, 480, 92160, 861557760, 300261256888320, 1835082185382168791040

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. A266177, A305568, A305571.

a(n) = A305568(n) * n!.

a(n) = A274806(n) - A305571(n).

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

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.

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.

A360221 Minimum number of intercalates in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 0, 0, 0, 2, 0

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An intercalate is a 2 X 2 subsquare of a Latin square.
An orthogonal diagonal Latin squares is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) <= 1, a(11) = 0, a(12) <= 4, a(13) = 0. - Eduard I. Vatutin, added Jan 30 2023, updated Sep 24 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, A307163, A354050, A360223.

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

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A287695 Maximum number of diagonal Latin squares with the first row in ascending order that can be orthogonal to a given diagonal Latin square of order n.

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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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A287762 Number of self-orthogonal diagonal Latin squares of order n.

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

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A305569 Number of bachelor diagonal Latin squares of order n.

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

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

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A360221 Minimum number of intercalates in an orthogonal diagonal Latin square of order n.

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