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

0, 2, 64, 97920

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) = 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
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
.
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   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. A287649, A339641, A340186.

Sequence in context: A348214 A092238 A228252 * A337651 A287649 A229352

Adjacent sequences: A339302 A339303 A339304 * A339306 A339307 A339308

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Dec 24 2020

Status

approved