OFFSET
1,2
COMMENTS
A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square. Diagonal Latin squares of this type have interesting properties, for example, a large number of transversals.
Plain symmetry diagonal Latin squares do not exist for odd orders, so a(2n+1)=0.
REFERENCES
J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, 1992, Vol. 139, pp. 43-49.
LINKS
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)
Eduard I. Vatutin, Enumeration of the Brown's diagonal Latin squares of orders 1-9 (in Russian).
FORMULA
a(n) = A340186(n) / n!. - Eduard I. Vatutin, Jan 08 2021
EXAMPLE
The diagonal Latin square
.
0 1 2 3 4 5 6 7 8 9
1 2 3 4 0 9 5 6 7 8
4 0 1 7 3 6 2 8 9 5
8 7 6 5 9 0 4 3 2 1
7 6 5 0 8 1 9 4 3 2
9 8 7 6 5 4 3 2 1 0
5 9 8 2 6 3 7 1 0 4
3 5 0 8 7 2 1 9 4 6
2 3 4 9 1 8 0 5 6 7
6 4 9 1 2 7 8 0 5 3
.
is a Brown's square since it is horizontally symmetric (see A287649) and its rows form row-inverse pairs:
.
0 1 2 3 4 5 6 7 8 9 . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 1 2 3 4 0 9 5 6 7 8 . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 4 0 1 7 3 6 2 8 9 5
. . . . . . . . . . 8 7 6 5 9 0 4 3 2 1 . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 8 7 6 5 4 3 2 1 0 . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 5 9 8 2 6 3 7 1 0 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
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7 6 5 0 8 1 9 4 3 2 . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 3 5 0 8 7 2 1 9 4 6
2 3 4 9 1 8 0 5 6 7 . . . . . . . . . .
. . . . . . . . . . 6 4 9 1 2 7 8 0 5 3
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Eduard I. Vatutin, Dec 24 2020
STATUS
approved