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A293778 Number of centrally symmetric diagonal Latin squares of order n. 4
1, 0, 0, 48, 960, 0, 14192640, 5449973760, 118753937326080 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Centrally symmetric diagon 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] (numbering of rows and columns from 0 to n-1).
It seems that a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist) and for n=2 (mod 4).
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A292517(n) <= a(4n). - Eduard I. Vatutin, May 26 2021
LINKS
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Properties of central symmetry for diagonal Latin squares, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
FORMULA
a(n) = A293777(n) * n!.
EXAMPLE
0 1 2 3 4 5 6 7 8
6 3 0 2 7 8 1 4 5
3 2 1 8 6 7 0 5 4
7 8 6 5 1 3 4 0 2
8 6 4 7 2 0 5 3 1
2 7 5 6 8 4 3 1 0
5 4 7 0 3 1 8 2 6
4 5 8 1 0 2 7 6 3
1 0 3 4 5 6 2 8 7
CROSSREFS
Sequence in context: A233177 A014703 A233674 * A089903 A274806 A292045
KEYWORD
nonn,more,hard
AUTHOR
Eduard I. Vatutin, Oct 16 2017
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)