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Number of extended self-orthogonal diagonal Latin squares of order n with the first row in ascending order.
3

%I #38 Jan 30 2023 08:46:53

%S 1,0,0,2,4,0,256,4608

%N Number of extended self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

%C A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

%C A333367(n) <= A287761(n) <= a(n) <= A305570(n). - _Eduard I. Vatutin_, Jun 07 2020

%C a(10) >= 510566400. - _Eduard I. Vatutin_, Jul 10 2020

%H E. I. Vatutin, <a href="https://vk.com/wall162891802_924">Discussion about properties of diagonal Latin squares</a> (in Russian).

%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1134">About the lower bound of number of ESODLS of order 10</a> (in Russian).

%H E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_esodls_1_to_8.zip">List of all main classes of extended self-orthogonal diagonal Latin squares of orders 1-8</a>.

%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 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>

%F From _Eduard I. Vatutin_, Feb 25 2020: (Start)

%F a(n) = A287761(n) for 1 <= n <= 6.

%F a(n) = 4*A287761(n) for 7 <= n <= 8. (End)

%F a(10) = A309210(10)*A299784(10) because no DSODLS exist for order n=10 and no ESODLS of order n=10 have generalized M-symmetries (automorphisms). - _Eduard I. Vatutin_, Jul 10 2020

%e The diagonal Latin square

%e 0 1 2 3 4 5 6 7 8 9

%e 1 2 0 4 5 7 9 8 6 3

%e 5 0 1 6 3 9 8 2 4 7

%e 9 3 5 8 2 1 7 4 0 6

%e 4 6 3 5 7 8 0 9 2 1

%e 8 4 6 9 1 3 2 5 7 0

%e 7 8 9 0 6 4 5 1 3 2

%e 2 9 4 7 8 0 3 6 1 5

%e 6 5 7 1 0 2 4 3 9 8

%e 3 7 8 2 9 6 1 0 5 4

%e has orthogonal diagonal Latin square

%e 0 1 2 3 4 5 6 7 8 9

%e 3 5 9 8 6 2 0 1 4 7

%e 4 3 8 7 2 1 9 0 5 6

%e 6 9 3 4 8 0 1 2 7 5

%e 7 2 0 1 9 3 5 8 6 4

%e 2 0 1 5 7 6 4 9 3 8

%e 8 6 4 2 0 9 7 5 1 3

%e 1 7 6 0 5 4 8 3 9 2

%e 9 8 5 6 1 7 3 4 2 0

%e 5 4 7 9 3 8 2 6 0 1

%e from the same main class.

%Y Cf. A287761, A305570, A333367.

%K nonn,more,hard

%O 1,4

%A _Eduard I. Vatutin_, Aug 09 2019