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%I #182 Aug 08 2023 22:23:11
%S 1,0,0,8,3,32,7,8,68
%N Minimum number of transversals in a diagonal Latin square of order n.
%C From _Eduard I. Vatutin_, Sep 20 2020: (Start)
%C Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A287644(n) <= A090741(n).
%C A lower bound for odd n is A091323((n-1)/2) <= a(n). (End)
%C By definition, the main diagonal and antidiagonal of a diagonal Latin square are transversals, so a(n)>=2 for all n>=4 (the two diagonals are the same in the order 1 square and there are no diagonal Latin squares of orders 2 or 3). - _Eduard I. Vatutin_, Jun 13 2021
%C All cyclic diagonal Latin squares are diagonal Latin squares, so a(n) <= A348212((n-1)/2) for all orders n of which cyclic diagonal Latin squares exist. - _Eduard I. Vatutin_, Mar 25 2021
%C a(10) <= 144, a(11) <= 1721, a(12) <= 448, a(13) <= 43093, a(14) <= 65432, a(15) <= 215721, a(16) <= 7465984. - _Eduard I. Vatutin_, Mar 11 2021, updated Jul 10 2023
%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> (in Russian).
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1347">About the minimal and maximal number of transversals in diagonal Latin squares of order 9</a> (in Russian).
%H Eduard I. Vatutin, <a href="/A287645/a287645_17.txt">Best examples presently known</a>.
%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, Recognition — 2017 (2017), pp. 98-100 (in Russian)
%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. Vatutin, A. Belyshev, N. Nikitina, and M. 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>, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.
%H E. I. Vatutin, 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_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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, and A. I. Pykhtin, <a href="http://evatutin.narod.ru/evatutin_dls_heur_spectra_method_2.pdf">Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares</a>, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (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 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e From _Eduard I. Vatutin_, Apr 24 2021: (Start)
%e For example, diagonal Latin square
%e 0 1 2 3
%e 3 2 1 0
%e 1 0 3 2
%e 2 3 0 1
%e has 4 diagonal transversals (see A287648)
%e 0 . . . . 1 . . . . 2 . . . . 3
%e . . 1 . . . . 0 3 . . . . 2 . .
%e . . . 2 . . 3 . . 0 . . 1 . . .
%e . 3 . . 2 . . . . . . 1 . . 0 .
%e and 4 not diagonal transversals
%e 0 . . . . 1 . . . . 2 . . . . 3
%e . 2 . . 3 . . . . . . 0 . . 1 .
%e . . 3 . . . . 2 1 . . . . 0 . .
%e . . . 1 . . 0 . . 3 . . 2 . . .
%e total 8 transversals. (End)
%Y Cf. A091323, A287644, A287647, A287648, A344105.
%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
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