Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n, https://oeis.org/A354068 n=1, a(1)=1 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: trivial 0 n=2, a(2)=0 - n=3, a(3)=0 - n=4, a(4)=4 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 3 2 1 0 1 0 3 2 2 3 0 1 n=5, a(5)=5 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 4 2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2 n=6, a(6)=0 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX (for n=6 ODLS does not exist) - n=7, a(7)=8 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 4 5 6 1 2 4 5 6 3 0 6 4 5 2 0 1 3 5 3 6 4 1 0 2 4 6 1 0 3 2 5 3 5 0 1 2 6 4 2 0 3 6 5 4 1 n=8, a(8)=8 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 2 3 0 1 7 6 5 4 5 6 1 7 3 4 0 2 6 5 3 4 1 7 2 0 7 4 5 2 6 0 3 1 4 7 6 0 5 2 1 3 1 0 4 5 2 3 7 6 3 2 7 6 0 1 4 5 n=9, a(9)=14 Article: 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. Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: brute force using X-based fillings, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 1 2 3 8 6 4 7 0 5 5 4 6 0 7 8 3 1 2 7 3 1 5 2 6 0 8 4 8 7 4 6 1 2 5 3 0 3 0 5 4 8 7 1 2 6 4 6 7 2 3 0 8 5 1 6 5 8 1 0 3 2 4 7 2 8 0 7 5 1 4 6 3 n=10, a(10)<=60 Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: different generators of DLS, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 1 2 0 4 5 3 8 9 7 6 6 7 5 9 8 0 2 1 4 3 3 0 9 7 6 1 4 8 2 5 7 8 4 1 3 6 0 5 9 2 8 9 7 5 0 4 3 2 6 1 4 6 3 2 1 8 9 0 5 7 9 5 8 0 2 7 1 6 3 4 2 3 6 8 7 9 5 4 1 0 5 4 1 6 9 2 7 3 0 8 n=11, a(11)<=279 Announcement: https://vk.com/wall162891802_1709, Eduard I. Vatutin, Jul 27 2021 Way of finding: different generators of DLS, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 10 1 2 3 10 5 9 8 4 6 7 0 3 4 10 7 1 6 0 5 9 8 2 5 3 9 1 8 10 7 0 2 6 4 9 0 8 6 7 4 3 2 10 5 1 4 10 0 5 2 8 1 6 7 3 9 8 7 5 2 6 0 9 10 4 1 3 7 8 6 4 9 2 10 3 1 0 5 10 6 1 9 0 3 4 8 5 2 7 6 5 7 0 10 1 2 9 3 4 8 2 9 4 8 3 7 5 1 0 10 6 n=12, a(12)<=588 Announcement: Eduard I. Vatutin, before Jul 19 2022 Way of finding: search for ODLS within the huge combinatorial structure, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 10 11 2 3 0 4 5 1 10 6 7 11 8 9 3 7 6 11 10 2 9 1 0 5 4 8 8 11 5 7 1 9 2 10 4 6 0 3 10 4 11 5 2 3 8 9 6 0 7 1 4 5 9 1 3 11 0 8 10 2 6 7 9 8 4 0 6 10 1 5 11 7 3 2 11 9 10 8 7 6 5 4 3 1 2 0 5 10 8 2 0 4 7 11 9 3 1 6 6 2 3 10 11 7 4 0 1 8 9 5 7 6 1 9 8 0 11 3 2 10 5 4 1 0 7 6 9 8 3 2 5 4 11 10 n=13, a(13)<=9610 Announcement: Eduard I. Vatutin, before Jul 19 2022 Way of finding: random search for DLS, Euler-Parker method, DLX 0 1 2 3 4 5 6 7 8 9 10 11 12 2 3 0 4 5 11 8 6 9 10 12 7 1 10 5 12 1 0 9 4 2 6 7 8 3 11 8 6 3 5 10 7 1 11 12 4 9 0 2 7 8 1 9 11 10 2 12 3 0 4 6 5 12 9 7 6 1 8 0 5 2 3 11 10 4 9 11 5 12 7 2 10 3 0 6 1 4 8 11 10 4 2 3 1 7 9 5 12 0 8 6 1 12 11 7 2 6 3 10 4 8 5 9 0 3 7 8 11 6 4 5 0 10 1 2 12 9 4 2 9 0 12 3 11 8 7 5 6 1 10 5 0 6 10 8 12 9 4 1 11 7 2 3 6 4 10 8 9 0 12 1 11 2 3 5 7 Jul 19 2022