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%I #74 Aug 08 2023 22:22:56
%S 1,0,0,2,4,0,64,1152,224832,234255360
%N Number of self-orthogonal diagonal Latin squares of order n with the first row in ascending order.
%C A self-orthogonal diagonal Latin square is a diagonal Latin square orthogonal to its transpose.
%C A333367(n) <= a(n) <= A309598(n) <= A305570(n). - _Eduard I. Vatutin_, Apr 26 2020
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1102">About the number of SODLS of order 10, a(10) value is wrong </a> (in Russian).
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1136">About the number of SODLS of order 10, corrected value a(10)</a> (in Russian).
%H E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_sodls_1_to_10.zip">List of all main classes of self-orthogonal diagonal Latin squares of orders 1-10</a>.
%H E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_dls_spec_types_list.pdf">Special types of diagonal Latin squares</a>, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
%H E. I. Vatutin and A. D. Belyshev, <a href="http://evatutin.narod.ru/evatutin_sodls_and_dsodls_1_to_10.pdf">About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10</a>. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (in Russian)
%H E. Vatutin and A. Belyshev, <a href="https://www.springerprofessional.de/en/enumerating-the-orthogonal-diagonal-latin-squares-of-small-order/18659992">Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality</a>, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.
%H Harry White, <a href="http://budshaw.ca/SODLS.html">Self-orthogonal Diagonal Latin Squares. How many</a>.
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F a(n) = A287762(n)/n!.
%F From _Eduard I. Vatutin_, Mar 14 2020: (Start)
%F a(i) != A329685(i)*A299784(i)/2 for i=1..9 due to the existence of doubly self-orthogonal diagonal Latin square (DSODLS) and/or generalized symmetries (automorphisms) for some SODLS.
%F a(10) = A329685(10)*A299784(10)/2 because no DSODLS exist for order n=10 and no SODLS of order n=10 have generalized symmetries (automorphisms). (End)
%e 0 1 2 3 4 5 6 7 8 9
%e 5 2 0 9 7 8 1 4 6 3
%e 9 5 7 1 8 6 4 3 0 2
%e 7 8 6 4 9 2 5 1 3 0
%e 8 9 5 0 3 4 2 6 7 1
%e 3 6 9 5 2 1 7 0 4 8
%e 4 3 1 7 6 0 8 2 9 5
%e 6 7 8 2 5 3 0 9 1 4
%e 2 0 4 6 1 9 3 8 5 7
%e 1 4 3 8 0 7 9 5 2 6
%Y Cf. A160368, A287762, A329685.
%K nonn,more,hard
%O 1,4
%A _Eduard I. Vatutin_, May 31 2017
%E a(10) from _Eduard I. Vatutin_, Mar 14 2020
%E a(10) corrected by _Eduard I. Vatutin_, Apr 24 2020
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