A339305 Number of Brown's diagonal Latin squares of order 2n with the first row in order.

0, 2, 128, 97920, 956301312

Offset: 1

Author

Eduard I. Vatutin, Dec 24 2020

Keywords

• nonn
• more
• hard

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.

Examples

			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
		

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).
• Eduard I. Vatutin, Clarification for Brown's diagonal Latin squares for orders 6 and 8 (in Russian).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A287649, A339641, A340186, A379145.

Formula

a(n) = A340186(n) / (2*n)!. - Eduard I. Vatutin, Jan 08 2021

Extensions

a(3) corrected by Eduard I. Vatutin and Oleg Zaikin, Dec 16 2024

a(5) added by Oleg S. Zaikin and Eduard I. Vatutin, Apr 08 2025