cp's OEIS Frontend

A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

%I A340545 #30 Aug 08 2023 22:21:57
%S A340545 1,0,0,1,2,0,32,301,430090
%N A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.
%C A340545 A centrally symmetric diagonal Latin square is a square with one-to-one correspondence between elements within all pairs a[i, j] and a[n-1-i, n-1-j] (with numbering of rows and columns from 0 to n-1).
%C A340545 It seems that a(n)=0 for n==2 (mod 4).
%C A340545 Centrally symmetric Latin squares are Latin squares, so a(n) <= A287764(n).
%C A340545 The canonical form (CF) of a square is the lexicographically minimal item within the corresponding main class of diagonal Latin square.
%C A340545 Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A340550(n) <= a(n). - _Eduard I. Vatutin_, May 28 2021
%H A340545 E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_dls_centr_symm.pdf">Properties of central symmetry for diagonal Latin squares</a>, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
%H A340545 E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, <a href="https://jpit.az/uploads/article/az/2019_2/CENTRAL_SYMMETRY_PROPERTIES_FOR_DIAGONAL_LATIN_SQUARES.pdf">Central Symmetry Properties for Diagonal Latin Squares</a>, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
%H A340545 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 A340545 E. I. Vatutin, <a href="https://vk.com/wall162891802_1448">About the number of main classes of centrally symmetric diagonal Latin squares of orders 1-9</a> (in Russian).
%H A340545 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 A340545 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A340545 For n=4 there is a single CF:
%e A340545   0 1 2 3
%e A340545   2 3 0 1
%e A340545   3 2 1 0
%e A340545   1 0 3 2
%e A340545 so a(4)=1.
%e A340545 For n=5 there are two different CFs:
%e A340545   0 1 2 3 4   0 1 2 3 4
%e A340545   2 3 4 0 1   1 3 4 2 0
%e A340545   4 0 1 2 3   4 2 1 0 3
%e A340545   1 2 3 4 0   2 0 3 4 1
%e A340545   3 4 0 1 2   3 4 0 1 2
%e A340545 so a(5)=2.
%e A340545 Example of a centrally symmetric diagonal Latin square of order n=9:
%e A340545   0 1 2 3 4 5 6 7 8
%e A340545   6 3 0 2 7 8 1 4 5
%e A340545   3 2 1 8 6 7 0 5 4
%e A340545   7 8 6 5 1 3 4 0 2
%e A340545   8 6 4 7 2 0 5 3 1
%e A340545   2 7 5 6 8 4 3 1 0
%e A340545   5 4 7 0 3 1 8 2 6
%e A340545   4 5 8 1 0 2 7 6 3
%e A340545   1 0 3 4 5 6 2 8 7
%Y A340545 Cf. A293777, A293778, A287764, A340550.
%K A340545 nonn,more,hard
%O A340545 1,5
%A A340545 _Eduard I. Vatutin_, Jan 11 2021