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%I #108 Aug 08 2023 22:23:00
%S 2,12288,81217160478720,6101215007109090122576072540160
%N Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.
%C A doubly symmetric square has symmetries in both the horizontal and vertical planes.
%C The plane symmetry requires one-to-one correspondence between the values of elements a[i,j] and a[N+1-i,j] in a vertical plane, and between the values of elements a[i,j] and a[i,N+1-j] in a horizontal plane for 1 <= i,j <= N. - _Eduard I. Vatutin_, Alexey D. Belyshev, Oct 09 2017
%C Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4).
%C Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A293777(4n). - _Eduard I. Vatutin_, May 26 2021
%C a(n)/(A001147(n)*2^(n*(4*n-3))) is the number of 2n X 2n grids with two instances of each of 1..n on the main diagonal and in each row and column with the first row in nondescreasing order. - _Andrew Howroyd_, May 30 2021
%H A. D. Belyshev, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=89143#post89143">Proof that the order of a doubly symmetric diagonal Latin squares is a multiple of 4</a>, 2017 (in Russian)
%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=87577#post87577">Discussion about properties of diagonal Latin squares at forum.boinc.ru, value a(4) is wrong</a> (in Russian)
%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=89332#post89332">Discussion about properties of diagonal Latin squares at forum.boinc.ru, corrected value a(4)</a> (in Russian)
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_1_7_trans_and_symm.pdf">Estimating of combinatorial characteristics for diagonal Latin squares</a>, Recognition — 2017 (2017), pp. 98-100 (in Russian)
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, <a href="http://ceur-ws.org/Vol-1940/paper10.pdf">On Some Features of Symmetric Diagonal Latin Squares</a>, CEUR WS, vol. 1940 (2017), pp. 74-79.
%H Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, Vitaly S. Titov, <a href="https://doi.org/10.25045/jpit.v10.i2.01">Central symmetry properties for diagonal Latin squares</a>, Problems of Information Technology (2019) No. 2, 3-8.
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_symm.pdf">Investigation of the properties of symmetric diagonal Latin squares</a>, Proceedings of the 10th multiconference on control problems (2017), vol. 3, pp. 17-19 (in Russian).
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_ls_dls_symm_v2.pdf">Investigation of the properties of symmetric diagonal Latin squares. Working on errors</a>, Intellectual and Information Systems (2017), pp. 30-36 (in Russian).
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1635">On the interconnection between double and central symmetries in diagonal Latin squares</a> (in Russian).
%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 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%F a(n) = A292517(n) / (4n)!.
%e Doubly symmetric diagonal Latin square example:
%e 0 1 2 3 4 5 6 7
%e 3 2 7 6 1 0 5 4
%e 2 3 1 0 7 6 4 5
%e 6 7 5 4 3 2 0 1
%e 7 6 3 2 5 4 1 0
%e 4 5 0 1 6 7 2 3
%e 5 4 6 7 0 1 3 2
%e 1 0 4 5 2 3 7 6
%e Reflection of all rows is equivalent to the exchange of elements 0 and 7, 1 and 6, 2 and 5, 3 and 4; hence, this square is horizontally symmetric. Reflection of all columns is equivalent to the exchange of elements 0 and 1, 2 and 4, 3 and 5, 6 and 7; hence, the square is also vertically symmetric.
%e From _Andrew Howroyd_, May 30 2021: (Start)
%e a(2) = 4*3*1024 = 12288. The 4 base quarter square arrangements are:
%e 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
%e 1 2 1 2 1 2 2 1 2 2 1 1 2 2 1 1
%e 2 1 2 1 2 2 1 1 1 1 2 2 2 2 1 1
%e 2 2 1 1 2 1 1 2 2 2 1 1 1 1 2 2
%e (End)
%Y Cf. A001147, A003191, A287649, A292517, A293777, A340550.
%K nonn,more,hard
%O 1,1
%A _Eduard I. Vatutin_, May 29 2017
%E a(2) corrected by _Eduard I. Vatutin_, Alexey D. Belyshev, Oct 09 2017
%E Edited and a(3) from Alexey D. Belyshev added by _Max Alekseyev_, Aug 23 2018, Sep 07 2018
%E a(4) from _Andrew Howroyd_, May 31 2021