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Minimal size of a main class for diagonal Latin squares of order n with the first row in ascending order.
4

%I #65 Mar 23 2022 07:37:37

%S 1,0,0,2,4,32,32,96

%N Minimal size of a main class for diagonal Latin squares of order n with the first row in ascending order.

%C a(9) <= 48; a(10) <= 7680, a(11) <= 1536, a(12) <= 46080, a(13) <= 7680. - _Eduard I. Vatutin_, Oct 05 2020, updated May 30 2021

%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, and 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, and 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="http://evatutin.narod.ru/evatutin_ls_cyclic_main_classes.pdf">Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares</a>, Recognition — 2021, pp. 77-79. (in Russian)

%H Eduard I. Vatutin, <a href="/A299783/a299783_1.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) = A299785(n) / n!.

%F 0 <= a(n) <= A299784(n). - _Eduard I. Vatutin_, Jun 08 2020

%F From _Eduard I. Vatutin_, May 30 2021: (Start)

%F A299783(n) = A299784(n) for 1 <= n <= 5.

%F A299783(6)*3 = A299784(6).

%F A299783(7)*6 = A299784(7).

%F A299783(8)*16 = A299784(8).

%F A299783(9)*32 = A299784(9).

%F A299783(10)*2 = A299784(10).

%F A299783(11)*10 = A299784(11).

%F A299783(12)*4 = A299784(12).

%F A299783(13)*24 = A299784(13). (End)

%e From _Eduard I. Vatutin_, Oct 05 2020: (Start)

%e The following DLS of order 9 has a main class with cardinality 48:

%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:

%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. A274171, A287764, A299784, A299785, A299787.

%K nonn,more,hard

%O 1,4

%A _Eduard I. Vatutin_, Jan 21 2019