Minimum number of transversals in a diagonal Latin square of order n, https://oeis.org/A287645 n=1, a(1)=1 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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. http://ceur-ws.org/Vol-1973/paper01.pdf Way of finding: brute force 0 n=2, a(2)=0 - n=3, a(3)=0 - n=4, a(4)=8 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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. http://ceur-ws.org/Vol-1973/paper01.pdf Way of finding: brute force 0 1 2 3 3 2 1 0 1 0 3 2 2 3 0 1 n=5, a(5)=3 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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. http://ceur-ws.org/Vol-1973/paper01.pdf Way of finding: brute force 0 1 2 3 4 4 2 0 1 3 1 4 3 2 0 3 0 1 4 2 2 3 4 0 1 n=6, a(6)=32 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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. http://ceur-ws.org/Vol-1973/paper01.pdf Way of finding: brute force 0 1 2 3 4 5 4 2 5 0 3 1 3 5 1 2 0 4 5 3 0 4 1 2 2 4 3 1 5 0 1 0 4 5 2 3 n=7, a(7)=7 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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. http://ceur-ws.org/Vol-1973/paper01.pdf Way of finding: brute force 0 1 2 3 4 5 6 6 2 3 5 0 1 4 4 5 1 6 2 3 0 2 3 6 4 5 0 1 1 6 5 0 3 4 2 5 0 4 2 1 6 3 3 4 0 1 6 2 5 n=8, a(8)=8 Article: E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, 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. http://ceur-ws.org/Vol-1973/paper01.pdf Way of finding: brute force 0 1 2 3 4 5 6 7 1 2 0 4 3 7 5 6 4 7 5 1 6 0 2 3 3 5 7 6 1 2 0 4 5 4 6 2 7 1 3 0 7 6 4 0 5 3 1 2 2 3 1 7 0 6 4 5 6 0 3 5 2 4 7 1 n=9, a(9)=68 Announcement: https://vk.com/wall162891802_1347, Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Sep 17 2020 Way of finding: brute force using X-based fillings 0 1 2 3 4 5 6 7 8 4 8 5 2 1 7 3 0 6 2 5 1 7 0 6 4 8 3 3 2 6 4 8 1 7 5 0 5 3 7 6 2 0 8 1 4 1 6 8 5 7 3 0 4 2 7 4 3 0 6 8 5 2 1 8 7 0 1 3 4 2 6 5 6 0 4 8 5 2 1 3 7 n=10, a(10)<=144 Announcement: https://vk.com/wall162891802_1799, Eduard I. Vatutin, Nov 01 2021 Way of finding: random search in the neighborhood of central symmetry, expanding spectra using 1 intercalate/loop rotation 0 8 9 6 5 2 4 7 3 1 4 2 6 3 9 0 5 1 7 8 2 6 5 0 4 8 9 3 1 7 9 4 7 1 3 5 6 2 8 0 6 5 1 8 7 9 3 0 2 4 7 1 3 9 8 4 0 5 6 2 1 3 0 4 2 7 8 9 5 6 8 7 2 5 0 3 1 6 4 9 3 0 8 7 6 1 2 4 9 5 5 9 4 2 1 6 7 8 0 3 n=11, a(11)<=1721 Announcement: https://vk.com/wall162891802_1828, Eduard I. Vatutin, Dec 14 2021 Way of finding: random search in the neighborhood of central symmetry + neighborhood based improvement, expanding spectra using 1 intercalate/loop rotation 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 6 9 10 8 7 0 5 3 10 8 7 0 4 1 9 5 2 6 8 5 9 10 2 1 3 4 6 7 0 5 3 4 6 9 7 8 1 0 10 2 2 6 10 8 7 3 0 5 9 1 4 4 8 7 0 1 6 5 10 2 3 9 7 9 5 2 3 0 4 6 10 8 1 10 4 6 9 8 2 7 0 1 5 3 6 7 0 1 5 10 9 2 3 4 8 9 0 1 5 10 8 2 3 4 6 7 n=12, a(12)<=448 Announcement: https://vk.com/wall162891802_1916, Eduard I. Vatutin, Jan 29 2022 Way of finding: random search in the neighborhood of different types of DLS, combinatorial structures ODLS CFs analysis, expanding transversals and diagonal transversals spectra using rotating of one intercalate/loop 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 0 8 10 11 9 7 4 6 5 6 11 9 4 1 8 5 3 2 0 7 10 8 7 6 11 10 9 2 1 4 5 0 3 7 9 11 5 6 0 3 2 1 10 8 4 2 3 5 10 7 4 9 11 6 8 1 0 11 5 0 8 2 7 10 4 9 1 3 6 10 6 1 2 5 3 7 8 0 11 4 9 5 10 4 1 11 2 8 0 3 6 9 7 9 0 10 6 3 1 4 5 11 7 2 8 4 8 7 9 0 11 1 6 10 3 5 2 3 4 8 7 9 6 0 10 5 2 11 1 n=13, a(13)<=43093 Announcement: https://vk.com/wall162891802_2161, Eduard I. Vatutin, Jan 09 2023 Way of finding: expanding transversals and diagonal transversals spectra using 1 intercalate/loop rotation, distributed diagonalizing 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 0 4 3 10 8 11 6 12 5 7 9 8 12 9 10 5 2 4 0 3 11 6 1 7 11 6 7 8 0 4 10 9 2 1 3 12 5 2 5 10 11 7 1 12 3 9 8 0 4 6 9 7 3 12 6 11 1 2 5 0 4 8 10 12 9 5 2 10 6 3 1 4 7 8 0 11 3 4 11 9 12 8 0 6 1 5 7 10 2 5 10 1 7 11 0 9 4 12 6 2 3 8 4 3 12 5 9 7 2 8 0 10 11 6 1 7 11 4 6 8 3 5 12 10 2 1 9 0 6 0 8 1 2 12 7 10 11 4 9 5 3 10 8 6 0 1 9 11 5 7 3 12 2 4 n=14, a(14)<=25720 Announcement: https://vk.com/wall162891802_2490, Eduard I. Vatutin, Aug 29 2023 Way of finding: from diagonal transversals spectrum 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 0 13 6 8 4 9 12 5 7 11 10 13 6 8 11 3 0 5 2 10 1 7 12 9 4 11 13 0 4 9 7 1 8 3 10 2 6 5 12 8 11 7 10 5 9 12 0 2 3 1 13 4 6 2 3 10 8 6 12 11 5 7 4 9 0 13 1 7 10 1 2 12 4 13 9 0 5 3 8 6 11 5 9 12 1 7 10 4 11 6 8 13 3 2 0 6 12 11 13 10 8 9 3 1 2 0 4 7 5 12 4 13 6 1 2 3 10 11 7 8 5 0 9 9 7 4 5 0 1 2 13 12 11 6 10 3 8 4 5 6 12 2 3 10 1 13 0 11 9 8 7 3 0 5 9 8 11 7 6 4 13 12 1 10 2 10 8 9 7 11 13 0 12 5 6 4 2 1 3 n=15, a(15)<=215721 Announcement: https://vk.com/wall162891802_1977, Eduard I. Vatutin, Apr 01 2022 Way of finding: expanding spectrum of diagonal transversals using intercalate/loop rotation neighborhoods 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 0 4 5 3 7 9 14 6 13 10 8 11 12 10 13 11 14 7 9 2 0 3 1 8 12 5 6 4 9 14 7 8 12 6 4 5 13 3 1 0 10 2 11 4 5 3 10 13 11 8 6 0 12 7 14 2 9 1 8 6 4 2 0 10 11 12 1 13 3 5 9 14 7 5 3 6 13 11 12 14 4 10 8 9 7 0 1 2 2 0 10 5 3 14 9 1 4 7 11 13 6 12 8 7 9 1 6 8 4 3 14 12 5 0 2 11 10 13 14 7 9 12 6 8 5 3 11 4 2 1 13 0 10 12 8 14 1 2 0 13 11 9 10 5 4 7 3 6 11 10 13 9 14 7 1 2 5 0 12 6 4 8 3 13 11 12 7 9 1 0 10 6 2 14 8 3 4 5 3 4 5 11 10 13 12 8 2 14 6 9 1 7 0 6 12 8 0 1 2 10 13 7 11 4 3 14 5 9 n=16, a(16)<=7465984 Announcement: https://vk.com/wall162891802_1903, Eduard I. Vatutin, Jan 23 2022 Way of finding: random Brown DLS, expanding spectrum of diagonal transversals using intercalate/loop rotation neighborhoods 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 0 4 5 11 7 9 14 10 13 12 15 6 3 8 8 13 1 10 12 3 0 6 7 15 11 5 9 14 2 4 5 12 4 9 2 0 11 10 3 13 15 6 7 8 1 14 13 8 14 5 15 9 1 2 0 12 4 10 3 7 6 11 11 4 15 0 14 6 10 3 5 1 9 7 2 12 8 13 10 15 9 2 13 7 3 5 4 6 0 8 14 11 12 1 4 10 3 1 8 14 9 12 2 11 7 13 6 15 5 0 6 14 13 15 9 10 8 0 11 2 5 3 4 1 7 12 14 7 8 11 3 15 13 1 6 4 12 9 10 2 0 5 2 6 7 12 10 4 14 13 1 5 8 15 11 0 9 3 3 9 10 7 1 8 15 11 12 0 6 14 13 5 4 2 7 0 6 8 11 12 2 14 13 3 1 4 5 9 15 10 9 3 5 14 0 13 12 4 15 7 2 1 8 10 11 6 15 11 12 6 7 2 5 8 10 14 3 0 1 4 13 9 12 5 11 13 6 1 4 15 9 8 14 2 0 3 10 7 Aug 29 2023