A339641 Number of main classes of Brown's diagonal Latin squares of order 2n.

0, 1, 2, 173, 124528

Offset

1,3

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..5.

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, Lists of canonical forms (4 <= N <= 8).

Index entries for sequences related to Latin squares and rectangles.

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, A339305, A340186.

Sequence in context: A262728 A139935 A281958 * A386618 A172231 A360478

Adjacent sequences: A339638 A339639 A339640 * A339642 A339643 A339644

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Dec 24 2020

Extensions

a(5) added by Eduard I. Vatutin from Oleg S. Zaikin, Mar 30 2025

Status

approved