A340186 Number of Brown's diagonal Latin squares of order 2n.

0, 48, 184320, 3948134400

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

Table of n, a(n) for n=1..4.

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).

Index entries for sequences related to Latin squares and rectangles.

Formula

a(n) = A339305(n) * n!.

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
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
.
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   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

Cf. A339305, A339641.

Sequence in context: A081262 A238001 A228143 * A008704 A037947 A282596

Adjacent sequences: A340183 A340184 A340185 * A340187 A340188 A340189

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Dec 31 2020

Extensions

a(3) corrected by Eduard I. Vatutin and Oleg Zaikin, Jan 12 2025

Status

approved