|
%I #129 Jan 09 2024 11:03:06
%S 1,0,0,8,15,32,133,384,2241
%N Maximum number of transversals in a diagonal Latin square of order n.
%C Same as the maximum number of transversals in a Latin square of order n except n = 3.
%C a(10) >= 5504 from Parker and Brown.
%C 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
%C 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
%C 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
%C 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
%D 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.
%D E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), pp. 73-81.
%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=87577#post87577">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a>.
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1347">About the minimal and maximal number of transversals in a diagonal Latin squares of order 9</a> (in Russian).
%H Eduard I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_ls_cyclic_main_classes.pdf">Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares</a>, Recognition — 2021, pp. 77-79. (in Russian)
%H Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, <a href="https://doi.org/10.1007/978-3-030-66895-2_9">Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10</a>, 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.
%H Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, <a href="https://doi.org/10.1007/978-3-031-49435-2_4">Diagonalization and Canonization of Latin Squares</a>, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.
%H E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_1_7_trans_and_symm.pdf">Estimating of combinatorial characteristics for diagonal Latin squares</a>, Recognition — 2017 (2017), pp. 98-100 (in Russian).
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, <a href="https://doi.org/10.25045/jpit.v10.i2.01">Central symmetry properties for diagonal Latin squares</a>, Problems of Information Technology (2019) No. 2, 3-8.
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, <a href="http://ceur-ws.org/Vol-1973/paper01.pdf">Enumerating the Transversals for Diagonal Latin Squares of Small Order</a>. 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.
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, <a href="https://doi.org/10.1515/eng-2017-0052">Using Volunteer Computing to Study Some Features of Diagonal Latin Squares</a>. Open Engineering. Vol. 7. Iss. 1. 2017, pp. 453-460. DOI: 10.1515/eng-2017-0052
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, and V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_dls_trans_enum.pdf">Estimating the Number of Transversals for Diagonal Latin Squares of Small Order</a>, Telecommunications. 2018. No. 1, pp. 12-21 (in Russian).
%H Eduard I. Vatutin, Natalia N. Nikitina, and Maxim O. Manzuk, <a href="https://vk.com/wall162891802_1485">First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch</a> (in Russian).
%H E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, <a href="http://evatutin.narod.ru/evatutin_spectra_t_dt_i_o_small_orders_thesis.pdf">On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order</a>, Intellectual and Information Systems (Intellect - 2021). Tula, 2021, pp. 7-17. (in Russian)
%H 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, <a href="http://evatutin.narod.ru/evatutin_spectra_t_dt_i_o_high_orders_1.pdf">Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9</a> (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022, pp. 314-315.
%H Eduard I. Vatutin, <a href="/A287644/a287644_3.txt">Proving list (best known examples)</a>.
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%Y Cf. A090741, A287645, A287647, A287648, A344105, A350585, A357514.
%K nonn,more,hard
%O 1,4
%A _Eduard I. Vatutin_, May 29 2017
%E a(8) added by _Eduard I. Vatutin_, Oct 29 2017
%E a(9) added by _Eduard I. Vatutin_, Sep 20 2020
|
