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A287647
Minimum number of diagonal transversals in a diagonal Latin square of order n.
10
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) <= 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 Oct 21 2024
LINKS
E. I. Vatutin, Discussion about properties of diagonal Latin squares, forum.boinc.ru (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).
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.
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
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