|
%I #67 May 31 2021 04:02:16
%S 1,0,0,48,480,23040,161280,3870720
%N Minimum size of a main class for diagonal Latin squares of order n.
%C 0 <= a(n) <= A299787(n). - _Eduard I. Vatutin_, Jun 08 2020
%C a(9) <= 17418240; a(10) <= 27869184000. - _Eduard I. Vatutin_, Oct 05 2020
%C a(11) <= 61312204800, a(12) <= 22072393728000, a(13) <= 47823519744000. - _Eduard I. Vatutin_, May 31 2021
%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_1234">About the upper bound of the minimal size of main class for diagonal Latin squares of order 9</a> (in Russian).
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1233">About the upper bound of the minimal size of main class for diagonal Latin squares of order 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 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="/A299785/a299785.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) = A299783(n) * n!.
%F From _Eduard I. Vatutin_, May 31 2021: (Start)
%F a(n) = A299787(n) for 1 <= n <= 5.
%F a(6) = A299787(6)/3.
%F a(7) = A299787(7)/6.
%F a(8) = A299787(8)/16.
%F a(9) = A299787(9)/32.
%F a(10) = A299787(10)/2.
%F a(11) = A299787(11)/10.
%F a(12) = A299787(12)/4.
%F a(13) = A299787(13)/24. (End)
%e From _Eduard I. Vatutin_, Oct 05 2020: (Start)
%e The following DLS of order 9 has a main class with cardinality 48*9! = 17418240:
%e 0 1 2 3 4 5 6 7 8
%e 2 4 3 0 7 6 8 1 5
%e 6 2 8 5 3 4 7 0 1
%e 4 6 7 1 8 2 3 5 0
%e 1 5 4 7 6 0 2 8 3
%e 7 8 1 4 5 3 0 6 2
%e 3 7 0 2 1 8 5 4 6
%e 8 3 5 6 0 7 1 2 4
%e 5 0 6 8 2 1 4 3 7
%e The following DLS of order 10 has a main class with cardinality 7680*10! = 27869184000:
%e 0 1 2 3 4 5 6 7 8 9
%e 1 2 0 4 3 6 5 9 7 8
%e 2 0 3 5 8 1 4 6 9 7
%e 4 6 9 7 1 8 2 0 3 5
%e 9 7 8 6 5 4 3 1 2 0
%e 3 4 7 8 0 9 1 2 5 6
%e 6 9 4 1 7 2 8 5 0 3
%e 7 8 5 0 6 3 9 4 1 2
%e 5 3 1 9 2 7 0 8 6 4
%e 8 5 6 2 9 0 7 3 4 1
%e (End)
%Y Cf. A287764, A299783, A299784, A299787.
%K nonn,more,hard
%O 1,4
%A _Eduard I. Vatutin_, Jan 21 2019
| 