A287645 -id:A287645 - cp's OEIS frontend

A287648 Maximum number of diagonal transversals in a diagonal Latin square of order n.

1, 0, 0, 4, 5, 6, 27, 120, 333

Offset: 1

Eduard I. Vatutin, May 29 2017

From Eduard I. Vatutin, Oct 04 2020: (Start)
A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element.
A diagonal transversal is a transversal that includes exactly one element from the main diagonal and exactly one from the antidiagonal. For squares of odd orders, these elements can coincide at the intersection of the diagonals. (End)
A007016 is an upper bound for the number of diagonal transversals in a Latin square: A287647(n) <= a(n) <= A007016(n). - Eduard I. Vatutin, Jan 02 2020
a(11) >= 4828, a(12) >= 24901, a(13) >= 131106, a(14) >= 364596, a(15) >= 389318. - Natalia Makarova, Tomáš Brada, Harry White, Oct 04 2020
a(16) >= 32172800, a(18) >= 280308432. - Natalia Makarova, Tomáš Brada, Dec 25 2020
a(12) >= 28496. - Natalia Makarova, Harry White, Jan 23 2021
a(14) >= 380718, a(20) >= 90010806304, a(21) >= 51162162017, a(22) >= 3227747329246. The number of D-transversals for orders 20 - 22 was calculated by a volunteer. - Natalia Makarova, Tomáš Brada, Harry White, Mar 17 2021
All cyclic diagonal Latin squares (see A338562) are diagonal Latin squares, so A342997((n-1)/2) <= a(n). - Eduard I. Vatutin, Apr 26 2021
a(14) >= 383578, a(15) >= 398974. - Natalia Makarova, Tomáš Brada, Jan 13 2022
a(10) >= 890, a(12) >= 30192, a(14) >= 490218, a(15) >= 4620434, a(17) >= 204995269, a(18) >= 281593874, a(19) >= 11254190082. - Eduard I. Vatutin, Jul 22 2020, updated Mar 01 2025
For most orders n, at least one diagonal Latin square with the maximal number of diagonal transversals has an orthogonal mate and a(n) = A360220(n). Known exceptions: n=6 and n=10. - Eduard I. Vatutin, Feb 17 2023

			For example, the diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals:
  0 . . .    . 1 . .    . . 2 .    . . . 3
  . . 1 .    . . . 0    3 . . .    . 2 . .
  . . . 2    . . 3 .    . 0 . .    1 . . .
  . 3 . .    2 . . .    . . . 1    . . 0 .
In addition there are 4 other transversals that are not diagonal transversals and are therefore not included here.
From _Natalia Makarova_, Oct 04 2020: (Start)
The following DLS of order 14 has 364596 diagonal transversals:
   0  7  6 11  9  3  4  5  2 12 13  8 10  1
   6  1 11  5 10 12  2  3  9  7  4 13  0  8
   5 11  2 12  8  1  7 10  0  6  9  3 13  4
  13  6  5  3  1 10  9 12  7  0  2  4  8 11
  12  3 10  1  4 13  8  6 11  5  0  7  2  9
  10 12  1  8  2  5 11 13  4  3  6  0  9  7
   9  2  7  0  5 11  6  8 13  4  1 10  3 12
   4 13  3  9  6  0 10  7  1  8 12  2 11  5
   2  4  9 10 11  6  1  0  8 13  7 12  5  3
   1 10  8 13 12  2  5  4  3  9 11  6  7  0
   3  5 12  7 13  8  0  1  6 11 10  9  4  2
   8  0 13  4  7  9  3  2 12 10  5 11  1  6
   7  9  0  6  3  4 13 11  5  2  8  1 12 10
  11  8  4  2  0  7 12  9 10  1  3  5  6 13
(End)

J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, and W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics. 1992. Vol. 139. pp. 43-49.

Tomáš Brada, Top 10 CF-ODLK with most orthogonal mates
Natalia Makarova, Most perfect diagonal Latin square of order 9 with 333 diagonal transversals
Natalia Makarova, ODLS of order n>10
Natalia Makarova, DLS with maximum of D-transversals
Natalia Makarova, DLS of orders n = 11 - 22 with known maximum of D-transversals
Natalia Makarova, Spectrum by D-transversals for the 14th order DLS
Natalia Makarova, Spectrum by D-transversals for the 15th order DLS
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition - 2021, pp. 77-79. (in Russian)
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, Diagonalization and Canonization of Latin Squares, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian).
Eduard I. Vatutin, Stepan E. Kochemazov, Oleg 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, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order, CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares, Open Engineering. Vol. 7. Iss. 1. 2017. pp. 453-460. DOI: 10.1515/eng-2017-0052.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1. pp. 12-21 (in Russian).
E. I. Vatutin, About the upper bound of number of diagonal transversals for diagonal Latin squares of order 10 (in Russian).
E. I. Vatutin, About the upper bound of number of diagonal transversals for diagonal Latin squares of order 9 (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, 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.
Eduard I. Vatutin, Best known examples.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A287644, A287645, A287647, A342997, A345370.

a(8) added by Eduard I. Vatutin, Oct 29 2017

a(9) added by Eduard I. Vatutin, Dec 08 2020

A287647 Minimum number of diagonal transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 1, 2, 0, 0, 0

Offset: 1

Eduard I. Vatutin, May 29 2017

A007016 is an upper bound for the number of diagonal transversals in a Latin square: a(n) <= A287648(n) <= A007016(n). - Eduard I. Vatutin, Jan 02 2020
From Eduard I. Vatutin, Apr 26 2021: (Start)
A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element.
A diagonal transversal is a transversal that includes exactly one element from the main diagonal and exactly one from the antidiagonal. For squares of odd orders, these elements can coincide at the intersection of the diagonals.
All cyclic diagonal Latin squares (see A338562) are diagonal Latin squares, so a(n) <= A342998((n-1)/2). (End)
a(10) <= 3, a(11) <= 43, a(12) = 0, a(13) <= 4756, a(14) <= 1446, a(15) <= 15510, a(16) <= 898988, a(17) <= 12058840, a(18) <= 82577875, a(19) <= 592174879, a(20) <= 4488686380. - Eduard I. Vatutin, Sep 26 2021, updated Jan 20 2025

			From _Eduard I. Vatutin_, Apr 26 2021: (Start)
For example, the diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals:
  0 . . .    . 1 . .    . . 2 .    . . . 3
  . . 1 .    . . . 0    3 . . .    . 2 . .
  . . . 2    . . 3 .    . 0 . .    1 . . .
  . 3 . .    2 . . .    . . . 1    . . 0 .
In addition there are 4 other transversals that are not diagonal transversals and are therefore not included here. (End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares, forum.boinc.ru (in Russian)
E. I. Vatutin, About the minimal number of diagonal transversals in a diagonal Latin squares of order 9 (in Russian)
Eduard I. Vatutin, Best known examples.
E. I. Vatutin, S. E. Kochemazov and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares. Open Engineering. Vol. 7. Iss. 1. 2017. pp. 453-460. DOI: 10.1515/eng-2017-0052
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1. pp. 12-21 (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, 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.
Index entries for sequences related to Latin squares and rectangles.

Cf. A287644, A287645, A287648, A342998, A345370.

a(8) added by Eduard I. Vatutin, Oct 29 2017

a(9) added by Eduard I. Vatutin, Sep 20 2020

A287644 Maximum number of transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 32, 133, 384, 2241

Offset: 1

Eduard I. Vatutin, May 29 2017

Same as the maximum number of transversals in a Latin square of order n except n = 3.
a(10) >= 5504 from Parker and Brown.
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= A357514(n) <= a(n) <= A090741(n). - Eduard I. Vatutin, added Sep 20 2020, updated Mar 03 2023
a(11) >= 37851, a(12) >= 198144, a(13) >= 1030367, a(14) >= 3477504, a(15) >= 36362925, a(16) >= 244744192, a(17) >= 1606008513, a(19) >= 87656896891, a(23) >= 452794797220965, a(25) >= 41609568918940625. - Eduard I. Vatutin, Mar 08 2020, updated Mar 10 2022
Also a(n) is the maximum number of transversals in an orthogonal diagonal Latin square of order n for all orders except n=6 where orthogonal diagonal Latin squares don't exist. - Eduard I. Vatutin, Jan 23 2022
All cyclic diagonal Latin squares are diagonal Latin squares, so A348212((n-1)/2) <= a(n) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021

J. W. Brown et al., Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, volume 139 (1992), pp. 43-49.
E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), pp. 73-81.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru.
E. I. Vatutin, About the minimal and maximal number of transversals in a diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition — 2021, pp. 77-79. (in Russian)
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer, Cham (2020), 127-146.
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, Diagonalization and Canonization of Latin Squares, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian).
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017, pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares. Open Engineering. Vol. 7. Iss. 1. 2017, pp. 453-460. DOI: 10.1515/eng-2017-0052
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1, pp. 12-21 (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, 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, 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.
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.
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A090741, A287645, A287647, A287648, A344105, A350585, A357514.

a(8) added by Eduard I. Vatutin, Oct 29 2017

a(9) added by Eduard I. Vatutin, Sep 20 2020

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

A357514 Minimum number of transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 0, 23, 16, 132

Offset: 1

Eduard I. Vatutin, Oct 01 2022

Orthogonal diagonal Latin squares is a diagonal Latin squares that have at least one orthogonal diagonal mate.
a(10) <= 652, a(11) <= 2091, a(12) <= 6240. - Eduard I. Vatutin, Oct 01 2022, updated Oct 21 2024
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= a(n) <= A287644(n) <= A090741(n). - Eduard I. Vatutin, Feb 17 2023

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, 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. A287644, A287645, A344105, A350585.

A348212 Number of transversals in a cyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 15, 133, 0, 37851, 1030367, 0, 1606008513, 87656896891, 0, 452794797220965, 41609568918940625

Offset: 1

Eduard I. Vatutin, Oct 07 2021

All cyclic diagonal Latin squares of order n have same number of transversals. A similar statement for diagonal transversals is not true (see A342998 and A342997).
All broken diagonals and antidiagonals of cyclic Latin squares are transversals, so a(n) >= 2*n for all n > 1 for which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 22 2022
All cyclic diagonal Latin squares are diagonal Latin squares, so A287645(2n+1) <= a(n) <= A287644(2n+1) for all orders in which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 23 2022

			A cyclic diagonal Latin square of order 5
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
has a(3)=15 transversals:
  0 . . . .   0 . . . .   . 1 . . .         . . . . 4
  . 3 . . .   . . . . 1   2 . . . .         . 3 . . .
  . . 1 . .   . . . 2 .   . . . . 3         . . . 2 .
  . . . 4 .   . . 3 . .   . . . 4 .         1 . . . .
  . . . . 2   . 4 . . .   . . 0 . .   ...   . . 0 . .

Eduard I. Vatutin, About the number of transversals in cyclic Latin and cyclic diagonal Latin squares (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006717, A011655, A287644, A287645, A338562, A342997, A342998.

a(n) = A006717(n) * A011655(n+1).

cp's OEIS Frontend

A287648 Maximum number of diagonal transversals in a diagonal Latin square of order n.

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A287647 Minimum number of diagonal transversals in a diagonal Latin square of order n.

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A287644 Maximum number of transversals in a diagonal Latin square of order n.

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

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A357514 Minimum number of transversals in an orthogonal diagonal Latin square of order n.

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A348212 Number of transversals in a cyclic diagonal Latin square of order 2n+1.

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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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