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%I #104 Aug 08 2023 22:23:20
%S 1,0,0,2,8,128,171200,7447587840,5056994653507584
%N Number of diagonal Latin squares of order n with the first row in order.
%C A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - _Andrew Howroyd_, Sep 29 2020
%H S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, <a href="https://arxiv.org/abs/1709.02599">Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order</a>, arXiv:1709.02599 [math.CO], 2017.
%H S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zaikin/zaikin3.html">Enumerating Diagonal Latin Squares of Order Up to 9</a>, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
%H M. O. Manzuk and N. N. Nikitina, <a href="https://rake.boincfast.ru/rakesearch/forum_thread.php?id=187">About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project</a>
%H Eduard I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=84942#post84942">a(9) value fixed after</a>
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1315">Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas</a> (in Russian)
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1330">Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas</a> (in Russian)
%H E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, <a href="http://evatutin.narod.ru/evatutin_dls_scf_gen.pdf">Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares</a>, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
%H E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, <a href="https://doi.org/10.1007/978-3-319-67035-5_9">Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9</a>, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
%H Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S.Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly 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, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_1_8_eng.pdf">Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares</a>, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
%H Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_1_8.pdf">The effect of filling cells order to the rate of generation of diagonal Latin squares</a>, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39 (in Russian).
%H E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_9.pdf">Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares</a>, Information technologies and mathematical modeling of systems (2016), pp. 154-157, (in Russian).
%H Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., and Titov V.S., <a href="http://ceur-ws.org/Vol-1787/486-490-paper-84.pdf">Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares</a>, CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
%H E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_dls_spec_types_list.pdf">Special types 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. 9-18. (in Russian)
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%F a(n) = A274806(n)/n!.
%e The a(4) = 2 diagonal Latin squares are:
%e 0 1 2 3 0 1 2 3
%e 2 3 0 1 3 2 1 0
%e 3 2 1 0 1 0 3 2
%e 1 0 3 2 2 3 0 1
%e .
%e The a(5) = 8 diagonal Latin squares are:
%e 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
%e 1 3 4 2 0 1 4 3 0 2 2 3 4 0 1 2 4 1 0 3
%e 4 2 1 0 3 3 2 1 4 0 4 0 1 2 3 4 0 3 2 1
%e 2 0 3 4 1 4 3 0 2 1 1 2 3 4 0 3 2 4 1 0
%e 3 4 0 1 2 2 0 4 1 3 3 4 0 1 2 1 3 0 4 2
%e .
%e 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
%e 3 4 0 1 2 3 4 1 2 0 4 2 0 1 3 4 2 3 0 1
%e 1 2 3 4 0 4 2 3 0 1 1 4 3 2 0 3 4 1 2 0
%e 4 0 1 2 3 2 0 4 1 3 3 0 1 4 2 1 3 0 4 2
%e 2 3 4 0 1 1 3 0 4 2 2 3 4 0 1 2 0 4 1 3
%Y Cf. A000315, A000479, A274806, A287764, A309283.
%K nonn,more,hard
%O 1,4
%A _Eduard I. Vatutin_, Jul 07 2016
%E a(9) added from Vatutin et al. (2016) by _Max Alekseyev_, Oct 05 2016
%E a(9) corrected by _Eduard I. Vatutin_, Oct 20 2016
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