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%I #19 Apr 15 2023 10:07:03
%S 1,0,2,3,20,67,596
%N Number of equivalence classes of X-based filling of diagonals in a diagonal Latin square of order 2n (or 2n+1).
%C Supplemental for A309283.
%C The number of solutions in an equivalence class with the main diagonal in ascending order is at most 4*2^n*n!. This maximum is only achieved for n >= 5. - _Andrew Howroyd_, Mar 27 2023
%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).
%F a(n) >= A000316(n) / (4*2^n*n!). - _Andrew Howroyd_, Mar 27 2023
%e From _Andrew Howroyd_, Mar 27 2023: (Start)
%e For n = 5, the following is an example solution in an equivalence class of maximum size. The second square shows the effect of swapping the two diagonals and renumbering so that the main diagonal is still in ascending order.
%e 0 . . . . . . . . 1 0 . . . . . . . . 1
%e . 1 . . . . . . 0 . . 1 . . . . . . 0 .
%e . . 2 . . . . 3 . . . . 2 . . . . 3 . .
%e . . . 3 . . 2 . . . . . . 3 . . 2 . . .
%e . . . . 4 6 . . . . . . . . 4 9 . . . .
%e . . . . 7 5 . . . . . . . . 6 5 . . . .
%e . . . 5 . . 6 . . . . . . 4 . . 6 . . .
%e . . 8 . . . . 7 . . . . 5 . . . . 7 . .
%e . 9 . . . . . . 8 . . 7 . . . . . . 8 .
%e 4 . . . . . . . . 9 8 . . . . . . . . 9
%e (End)
%Y Cf. A000316, A309283.
%K nonn,more,hard
%O 0,3
%A _Eduard I. Vatutin_, Oct 08 2020
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