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

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

Offset

1,4

Comments

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) <= 145, a(12) = 0, a(13) <= 4756, a(14) <= 5173, 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 Jul 10 2023

Links

Table of n, a(n) for n=1..9.

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 examples known.

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.

Example

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)

Crossrefs

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

Sequence in context: A357052 A121066 A343635 * A331749 A274438 A365387

Adjacent sequences: A287644 A287645 A287646 * A287648 A287649 A287650

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, May 29 2017

Extensions

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

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

Status

approved