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%I #22 Jun 13 2021 07:14:53
%S 0,0,0,12,12,51,151,924
%N Maximum number of nontrivial Latin subrectangles in a diagonal Latin square of order n.
%C A Latin subrectangle is an m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.
%C A nontrivial Latin subrectangle is an m X k Latin rectangle of a Latin square of order n, 1 < m < n, 1 < k < n.
%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=92687#post92687">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian).
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1322">About the minimum and maximum number of nontrivial Latin subrectangles in a diagonal Latin squares of order 8</a> (in Russian).
%H Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, <a href="https://doi.org/10.1007/978-3-030-66895-2_9">Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10</a>, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
%H Eduard I. Vatutin, <a href="/A307842/a307842.txt">Proving list (best known examples)</a>.
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e For example, the square
%e 0 1 2 3 4 5 6
%e 4 2 6 5 0 1 3
%e 3 6 1 0 5 2 4
%e 6 3 5 4 1 0 2
%e 1 5 3 2 6 4 0
%e 5 0 4 6 2 3 1
%e 2 4 0 1 3 6 5
%e has nontrivial Latin subrectangle
%e . . . . . . .
%e . . 6 5 0 1 3
%e . . . . . . .
%e . . . . . . .
%e . . . . . . .
%e . . . . . . .
%e . . 0 1 3 6 5
%e The total number of Latin subrectangles for this square is 2119 and the number of nontrivial Latin subrectangles is only 151.
%Y Cf. A307840, A307841.
%K nonn,more,hard
%O 1,4
%A _Eduard I. Vatutin_, May 01 2019
%E a(8) added by _Eduard I. Vatutin_, Oct 06 2020
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