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A340186 Number of Brown's diagonal Latin squares of order 2n. 2
0, 48, 92160, 3948134400 (list; graph; refs; listen; history; text; internal format)
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.

Eduard I. Vatutin, Enumeration of the Brown's diagonal Latin squares of orders 1-9 (in Russian).

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: A292516 A006070 A081262 * A238001 A228143 A008704

Adjacent sequences:  A340183 A340184 A340185 * A340187 A340188 A340189

KEYWORD

nonn,more,hard

AUTHOR

Eduard I. Vatutin, Dec 31 2020

STATUS

approved

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Last modified January 24 11:46 EST 2022. Contains 350536 sequences. (Running on oeis4.)