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%I #115 Jun 01 2021 03:09:10
%S 1,0,0,48,480,69120,967680,61931520,557383680,55738368000,
%T 613122048000,88289574912000,1147764473856000,224961836875776000,
%U 3374427553136640000
%N Maximum size of a main class for diagonal Latin squares of order n.
%C a(n) <= 2^m * m! * 4 * n!, where m = floor(n/2).
%C It seems that a(n) = 2^m * m! * 4 * n! for all n>6. - _Eduard I. Vatutin_, Jun 08 2020
%C 0 <= A299785(n) <= a(n). - _Eduard I. Vatutin_, Jul 06 2020
%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=92076#post92076">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian).
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1103">About the maximal size of main classes of diagonal Latin squares of orders 9 and 10</a> (in Russian).
%H E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, <a href="http://evatutin.narod.ru/evatutin_co_dls_cfs_cnt.pdf">Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing</a>, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
%H E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, <a href="https://doi.org/10.1007/978-3-030-05807-4_49">Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing</a>, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1106">About the maximal size of main class for diagonal Latin squares of orders 11-15</a> (in Russian).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1575">About the relationship between the minimal and maximal cardinality of main classes for diagonal Latin squares</a> (in Russian).
%H Eduard I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_dls_mc_max_card_9_to_15.pdf">Estimating the maximal size of main class for diagonal Latin squares of orders 9-15</a>, Medical-Ecological and Information Technologies - 2020, Part 2, 2020, pp. 57-62 (in Russian).
%H Eduard I. Vatutin, <a href="/A299787/a299787.txt">Proving list (best known examples)</a>.
%H <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F a(n) = A299784(n) * n!.
%F From _Eduard I. Vatutin_, May 31 2021: (Start)
%F a(n) = A299785(n) for 1 <= n <= 5.
%F a(6) = A299785(6)*3.
%F a(7) = A299785(7)*6.
%F a(8) = A299785(8)*16.
%F a(9) = A299785(9)*32.
%F a(10) = A299785(10)*2.
%F a(11) = A299785(11)*10.
%F a(12) = A299785(12)*4.
%F a(13) = A299785(13)*24. (End)
%e From _Eduard I. Vatutin_, May 31 2021: (Start)
%e The following DLS of order 9 has a main class with cardinality 1536*9! = 557383680:
%e 0 1 2 3 4 5 6 7 8
%e 1 2 0 4 8 6 5 3 7
%e 7 4 5 8 0 3 2 6 1
%e 5 8 7 6 1 0 3 2 4
%e 8 0 3 2 7 1 4 5 6
%e 3 7 8 5 6 4 1 0 2
%e 6 3 1 7 5 2 8 4 0
%e 2 6 4 0 3 8 7 1 5
%e 4 5 6 1 2 7 0 8 3
%e The following DLS of order 10 has a main class with cardinality 15360*10! = 55738368000:
%e 0 1 2 3 4 5 6 7 8 9
%e 1 2 0 4 5 3 9 8 6 7
%e 3 5 6 1 8 7 4 0 9 2
%e 9 4 7 8 3 2 1 6 0 5
%e 2 7 3 0 9 8 5 1 4 6
%e 6 8 5 9 2 4 7 3 1 0
%e 4 6 9 7 0 1 3 2 5 8
%e 7 0 4 6 1 9 8 5 2 3
%e 8 3 1 5 6 0 2 9 7 4
%e 5 9 8 2 7 6 0 4 3 1
%e (End)
%Y Cf. A287764, A299783, A299784, A299785.
%K nonn,more
%O 1,4
%A _Eduard I. Vatutin_, Jan 21 2019
%E a(9)-a(10) from _Eduard I. Vatutin_, Mar 15 2020
%E a(11)-a(15) from _Eduard I. Vatutin_, Jun 08 2020
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