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

%I

%S 0,48,92160,3948134400

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

%C 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.

%C Plain symmetry diagonal Latin squares are not exists for odd orders, so a(2n+1)=0.

%D 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.

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1471">Enumeration of the Brown's diagonal Latin squares of orders 1-9</a> (in Russian).

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

%e Diagonal Latin square

%e 0 1 2 3 4 5 6 7 8 9

%e 1 2 3 4 0 9 5 6 7 8

%e 4 0 1 7 3 6 2 8 9 5

%e 8 7 6 5 9 0 4 3 2 1

%e 7 6 5 0 8 1 9 4 3 2

%e 9 8 7 6 5 4 3 2 1 0

%e 5 9 8 2 6 3 7 1 0 4

%e 3 5 0 8 7 2 1 9 4 6

%e 2 3 4 9 1 8 0 5 6 7

%e 6 4 9 1 2 7 8 0 5 3

%e is a Brown's square due to it is horizontally symmetric (see A287649) and its rows forms row-inverse pairs:

%e 0 1 2 3 4 5 6 7 8 9 . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . 1 2 3 4 0 9 5 6 7 8 . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . . 4 0 1 7 3 6 2 8 9 5

%e . . . . . . . . . . 8 7 6 5 9 0 4 3 2 1 . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

%e 9 8 7 6 5 4 3 2 1 0 . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . . 5 9 8 2 6 3 7 1 0 4

%e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

%e .

%e . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . .

%e 7 6 5 0 8 1 9 4 3 2 . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . . . . . . . . .

%e . . . . . . . . . . 3 5 0 8 7 2 1 9 4 6

%e 2 3 4 9 1 8 0 5 6 7 . . . . . . . . . .

%e . . . . . . . . . . 6 4 9 1 2 7 8 0 5 3

%Y Cf. A339305, A339641.

%K nonn,more,hard,new

%O 1,2

%A _Eduard I. Vatutin_, Dec 31 2020

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Last modified January 18 00:11 EST 2021. Contains 340249 sequences. (Running on oeis4.)