A287764 -id:A287764 - cp's OEIS frontend

A274171 Number of diagonal Latin squares of order n with the first row in order.

1, 0, 0, 2, 8, 128, 171200, 7447587840, 5056994653507584

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Sep 29 2020

			The a(4) = 2 diagonal Latin squares are:
   0 1 2 3   0 1 2 3
   2 3 0 1   3 2 1 0
   3 2 1 0   1 0 3 2
   1 0 3 2   2 3 0 1
.
The a(5) = 8 diagonal Latin squares are:
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   1 3 4 2 0   1 4 3 0 2   2 3 4 0 1   2 4 1 0 3
   4 2 1 0 3   3 2 1 4 0   4 0 1 2 3   4 0 3 2 1
   2 0 3 4 1   4 3 0 2 1   1 2 3 4 0   3 2 4 1 0
   3 4 0 1 2   2 0 4 1 3   3 4 0 1 2   1 3 0 4 2
.
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   3 4 0 1 2   3 4 1 2 0   4 2 0 1 3   4 2 3 0 1
   1 2 3 4 0   4 2 3 0 1   1 4 3 2 0   3 4 1 2 0
   4 0 1 2 3   2 0 4 1 3   3 0 1 4 2   1 3 0 4 2
   2 3 4 0 1   1 3 0 4 2   2 3 4 0 1   2 0 4 1 3

S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.
S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
Eduard I. Vatutin, a(9) value fixed after
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
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.
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39 (in Russian).
E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157, (in Russian).
Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., and Titov V.S., Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
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)
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000479, A274806, A287764, A309283.

a(n) = A274806(n)/n!.

a(9) added from Vatutin et al. (2016) by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A274806 Number of diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 960, 92160, 862848000, 300286741708800, 1835082219864832081920

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Oct 05 2020

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian).
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian).
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
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.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39, (in Russian).
E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157, (in Russian).
Eduard I. Vatutin, a(9) value fixed
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
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)
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000479, A274171, A287764, A309283.

a(n) = A274171(n) * n!.

a(9) from Vatutin et al. (2016) added by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A344105 a(n) is the number of distinct numbers of transversals of order n diagonal Latin squares.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 32, 73, 406

Offset: 1

Eduard I. Vatutin, Jun 22 2021

a(n) <= A287644(n) - A287645(n) + 1.
a(n) <= A287764(n).
Diagonal Latin squares are a special case of Latin squares, so a(n) <= A309344(n).
a(10) >= 459, a(11) >= 6437, a(12) >= 23707, a(13) >= 75891, a(14) >= 290681. - Eduard I. Vatutin, Oct 29 2021, updated Mar 01 2025
For all spectra of even orders all known values included in them are divisible by 2. For all spectra of orders n=6, n=10 and n=14 (and probably for all n=4k+2) all known values included in the corresponding spectra are divisible by 4. This leads to the following hypothesis: a(2k) <= (A287644(2k) - A287645(2k) + 2)/2 and a(4k+2) <= (A287644(4k+2) - A287645(4k+2) + 4)/4, where w(n) = A287644(n) - A287645(n) + 1 is a width of corresponding spectra and (w(n)+1)/2 is done to round the result of the division up. - Eduard I. Vatutin, Mar 21 2022

			For n=7 the number of transversals that a diagonal Latin square of order 7 may have is 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 41, 43, 45, 47, 55, or 133. Since there are 32 distinct values, a(7)=32.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, and A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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.
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, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, and A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
Index entries for sequences related to Latin squares and rectangles.

Cf. A287644, A287645, A287764, A309344, A345370, A345760, A345761.

a(8) added by Eduard I. Vatutin, Jul 14 2021

a(9) added by Eduard I. Vatutin, Nov 20 2022

A345760 a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 21, 61, 64

Offset: 1

Eduard I. Vatutin, Jun 26 2021

a(n) <= A307164(n) - A307163(n) + 1.
a(n) <= A287764(n).
a(10) >= 98, a(11) >= 145, a(12) >= 259, a(13) >= 200, a(14) >= 362, a(15) >= 536, a(16) >= 792, a(17) >= 685, a(18) >= 535, a(19) >= 447, a(20) >= 1011, a(21) >= 747, a(22) >= 872, a(23) >= 885, a(24) >= 1610, a(25) >= 1677, a(26) >= 1266, a(27) >= 1337, a(28) >= 2795. - Eduard I. Vatutin, Oct 02 2021, updated Mar 02 2025

			For n=7 the number of intercalates that a diagonal Latin square of order 7 may have is 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26, or 30. Since there are 21 distinct values, a(7)=21.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A307163, A307164, A309344, A344105, A345370, A345761.

a(9) added by Eduard I. Vatutin, Oct 22 2022

A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 14, 47, 182

Offset: 1

Eduard I. Vatutin, Jun 16 2021

a(n) <= A287648(n) - A287647(n) + 1.
a(n) <= A287764(n).
Conjecture: a(12) = A287648(12) - A287647(12) + 1. - Natalia Makarova, Oct 26 2021
a(10) >= 736, a(11) >= 1344, a(12) >= 17693, a(13) >= 18241, a(14) >= 294053, a(15) >= 1958394, a(16) >= 13715. - Eduard I. Vatutin, Oct 29 2021, updated Mar 02 2025

			For n=7 the number of diagonal transversals that a diagonal Latin square of order 7 may have is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, or 27. Since there are 14 distinct values, a(7)=14.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, On the falsity of conjecture that spectra of diagonal transversals for diagonal Latin squares of order 12 is solid (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
E. I. Vatutin, Distributed diagonalization strategy for Latin squares, Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2023. pp. 309-311. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, 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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A309344, A344105, A345760, A345761, A349199.

a(8) added by Eduard I. Vatutin, Jul 15 2021

a(9) added by Eduard I. Vatutin, Oct 20 2022

A299784 Maximum 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, 96, 192, 1536, 1536, 15360, 15360, 184320, 184320, 2580480, 2580480

Offset: 1

Eduard I. Vatutin, Jan 21 2019

a(n) <= 2^m * m! * 4, where m = floor(n/2).
It seems that a(n) = 2^m * m! * 4 for all n > 6. - Eduard I. Vatutin, Jun 08 2020
0 <= A299783(n) <= a(n). - Eduard I. Vatutin, Jun 08 2020

			From _Eduard I. Vatutin_, May 30 2021: (Start)
The following DLS of order 9 has a main class with cardinality 1536:
  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:
  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. 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.
E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian).
E. I. Vatutin, About the maximal size of main class for diagonal Latin squares of orders 11-15 (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, 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.

a(n) = A299787(n) / n!.
From Eduard I. Vatutin, May 30 2021: (Start)
A299783(n) = a(n) for 1 <= n <= 5.
A299783(6)*3 = a(6).
A299783(7)*6 = a(7).
A299783(8)*16 = a(8).
A299783(9)*32 <= a(9).
A299783(10)*10 <= a(10).
A299783(11)*10 <= a(11).
A299783(12)*4 <= a(12).
A299783(13)*24 <= a(13). (End)

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

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

A340546 Number of main classes of diagonal Latin squares of order 2n that contain a one-plane symmetric square.

Original entry on oeis.org

0, 1, 2, 9717

Offset: 1

Eduard I. Vatutin, Jan 11 2021

A one-plane symmetric diagonal Latin square is a vertically or horizontally symmetric diagonal Latin square (see A296060). Such diagonal Latin squares do not exist for odd orders > 1.

			A horizontally symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 0 5 3 1
  5 4 3 2 1 0
  2 5 4 1 0 3
  3 0 1 4 5 2
  1 3 5 0 2 4
A vertically symmetric diagonal Latin square:
  0 1 2 3 4 5
  4 2 5 0 3 1
  3 5 1 2 0 4
  5 3 0 4 1 2
  2 4 3 1 5 0
  1 0 4 5 2 3
Both are one-plane symmetric diagonal Latin squares.

Eduard I. Vatutin, On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-8 (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)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287649, A287764, A292516, A293777, A293778, A296060, A296061, A340550.

Name clarified by Andrew Howroyd, Oct 22 2023

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

cp's OEIS Frontend

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

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