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

Author

Eduard I. Vatutin, May 29 2017

Keywords

• nonn
• more
• hard

Comments

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

Examples

			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)
		

References

• 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.

Links

• 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.

Crossrefs

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

Extensions

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

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