A299784 -id:A299784 - 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

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

A299783 Minimum size of a main class for diagonal Latin squares of order n with the first row in ascending order.

Original entry on oeis.org

1, 0, 0, 2, 4, 32, 32, 96

Offset: 1

Eduard I. Vatutin, Jan 21 2019

a(9) <= 48; a(10) <= 1536, a(11) <= 1536, a(12) <= 46080, a(13) <= 7680. - Eduard I. Vatutin, Oct 05 2020, updated Apr 08 2025

			From _Eduard I. Vatutin_, Oct 05 2020: (Start)
The following DLS of order 9 has a main class with cardinality 48:
  0 1 2 3 4 5 6 7 8
  2 4 3 0 7 6 8 1 5
  6 2 8 5 3 4 7 0 1
  4 6 7 1 8 2 3 5 0
  1 5 4 7 6 0 2 8 3
  7 8 1 4 5 3 0 6 2
  3 7 0 2 1 8 5 4 6
  8 3 5 6 0 7 1 2 4
  5 0 6 8 2 1 4 3 7
The following DLS of order 10 has a main class with cardinality 7680:
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 3 6 5 9 7 8
  2 0 3 5 8 1 4 6 9 7
  4 6 9 7 1 8 2 0 3 5
  9 7 8 6 5 4 3 1 2 0
  3 4 7 8 0 9 1 2 5 6
  6 9 4 1 7 2 8 5 0 3
  7 8 5 0 6 3 9 4 1 2
  5 3 1 9 2 7 0 8 6 4
  8 5 6 2 9 0 7 3 4 1
(End)

E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 9 (in Russian).
E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 10 (in Russian).
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, and N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, and N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
Eduard I. Vatutin, About the relationship between the minimal and maximal cardinality of main classes for diagonal Latin squares (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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274171, A287764, A299784, A299785, A299787.

a(n) = A299785(n) / n!.
0 <= a(n) <= A299784(n). - Eduard I. Vatutin, Jun 08 2020
From Eduard I. Vatutin, added May 30 2021, updated Apr 08 2025: (Start)
a(n) = A299784(n) for 1 <= n <= 5.
a(6)*3 = A299784(6).
a(7)*6 = A299784(7).
a(8)*16 = A299784(8).
a(9)*32 <= A299784(9).
a(10)*10 <= A299784(10).
a(11)*10 <= A299784(11).
a(12)*4 <= A299784(12).
a(13)*24 <= A299784(13). (End)

A299787 Maximum size of a main class for diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 480, 69120, 967680, 61931520, 557383680, 55738368000, 613122048000, 88289574912000, 1147764473856000, 224961836875776000, 3374427553136640000

Offset: 1

Eduard I. Vatutin, Jan 21 2019

a(n) <= 2^m * m! * 4 * n!, where m = floor(n/2).
It seems that a(n) = 2^m * m! * 4 * n! for all n>6. - Eduard I. Vatutin, Jun 08 2020
0 <= A299785(n) <= a(n). - Eduard I. Vatutin, Jul 06 2020

			From _Eduard I. Vatutin_, May 31 2021: (Start)
The following DLS of order 9 has a main class with cardinality 1536*9! = 557383680:
  0 1 2 3 4 5 6 7 8
  1 2 0 4 8 6 5 3 7
  7 4 5 8 0 3 2 6 1
  5 8 7 6 1 0 3 2 4
  8 0 3 2 7 1 4 5 6
  3 7 8 5 6 4 1 0 2
  6 3 1 7 5 2 8 4 0
  2 6 4 0 3 8 7 1 5
  4 5 6 1 2 7 0 8 3
The following DLS of order 10 has a main class with cardinality 15360*10! = 55738368000:
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 5 3 9 8 6 7
  3 5 6 1 8 7 4 0 9 2
  9 4 7 8 3 2 1 6 0 5
  2 7 3 0 9 8 5 1 4 6
  6 8 5 9 2 4 7 3 1 0
  4 6 9 7 0 1 3 2 5 8
  7 0 4 6 1 9 8 5 2 3
  8 3 1 5 6 0 2 9 7 4
  5 9 8 2 7 6 0 4 3 1
(End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the maximal size of main classes of diagonal Latin squares of orders 9 and 10 (in Russian).
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
E. I. Vatutin, About the maximal size of main class for diagonal Latin squares of orders 11-15 (in Russian).
Eduard I. Vatutin, About the relationship between the minimal and maximal cardinality of main classes for diagonal Latin squares (in Russian).
Eduard I. Vatutin, Estimating the maximal size of main class for diagonal Latin squares of orders 9-15, Medical-Ecological and Information Technologies - 2020, Part 2, 2020, pp. 57-62 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A299783, A299784, A299785.

a(n) = A299784(n) * n!.
From Eduard I. Vatutin, May 31 2021: (Start)
a(n) = A299785(n) for 1 <= n <= 5.
a(6) = A299785(6)*3.
a(7) = A299785(7)*6.
a(8) = A299785(8)*16.
a(9) = A299785(9)*32.
a(10) = A299785(10)*2.
a(11) = A299785(11)*10.
a(12) = A299785(12)*4.
a(13) = A299785(13)*24. (End)

a(9)-a(10) from Eduard I. Vatutin, Mar 15 2020

a(11)-a(15) from Eduard I. Vatutin, Jun 08 2020

A299785 Minimum size of a main class for diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 480, 23040, 161280, 3870720

Offset: 1

Eduard I. Vatutin, Jan 21 2019

0 <= a(n) <= A299787(n). - Eduard I. Vatutin, Jun 08 2020
a(9) <= 17418240; a(10) <= 27869184000. - Eduard I. Vatutin, Oct 05 2020
a(11) <= 61312204800, a(12) <= 22072393728000, a(13) <= 47823519744000. - Eduard I. Vatutin, May 31 2021

			From _Eduard I. Vatutin_, Oct 05 2020: (Start)
The following DLS of order 9 has a main class with cardinality 48*9! = 17418240:
  0 1 2 3 4 5 6 7 8
  2 4 3 0 7 6 8 1 5
  6 2 8 5 3 4 7 0 1
  4 6 7 1 8 2 3 5 0
  1 5 4 7 6 0 2 8 3
  7 8 1 4 5 3 0 6 2
  3 7 0 2 1 8 5 4 6
  8 3 5 6 0 7 1 2 4
  5 0 6 8 2 1 4 3 7
The following DLS of order 10 has a main class with cardinality 7680*10! = 27869184000:
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 3 6 5 9 7 8
  2 0 3 5 8 1 4 6 9 7
  4 6 9 7 1 8 2 0 3 5
  9 7 8 6 5 4 3 1 2 0
  3 4 7 8 0 9 1 2 5 6
  6 9 4 1 7 2 8 5 0 3
  7 8 5 0 6 3 9 4 1 2
  5 3 1 9 2 7 0 8 6 4
  8 5 6 2 9 0 7 3 4 1
(End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 9 (in Russian).
E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 10 (in Russian).
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
Eduard I. Vatutin, About the relationship between the minimal and maximal cardinality of main classes for diagonal Latin squares (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A299783, A299784, A299787.

a(n) = A299783(n) * n!.
From Eduard I. Vatutin, May 31 2021: (Start)
a(n) = A299787(n) for 1 <= n <= 5.
a(6) = A299787(6)/3.
a(7) = A299787(7)/6.
a(8) = A299787(8)/16.
a(9) = A299787(9)/32.
a(10) = A299787(10)/2.
a(11) = A299787(11)/10.
a(12) = A299787(12)/4.
a(13) = A299787(13)/24. (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A287761 Number of self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A299783 Minimum size of a main class for diagonal Latin squares of order n with the first row in ascending order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A299787 Maximum size of a main class for diagonal Latin squares of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A299785 Minimum size of a main class for diagonal Latin squares of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula