A342997 - OEIS

%I #23 May 05 2021 13:58:52

%S 1,0,5,27,0,4665,131106,0,204995269,11254190082

%N Maximum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

%C A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places (see A338562, A123565 and A341585).

%C Cyclic diagonal Latin squares do not exist for even n.

%C All cyclic diagonal Latin squares are diagonal Latin squares, so a((n-1)/2) <= A287648(n).

%C All diagonal transversals are transversals, so a(n) <= A006717(n).

%C A342998 <= a(n).

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1412">Enumerating the diagonal transversals for cyclic diagonal Latin squares of orders 1-19</a> (in Russian).

%H Eduard I. Vatutin, <a href="/A342997/a342997.txt">Proving list (best known examples)</a>.

%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.

%e For n=2 one of the best cyclic diagonal Latin squares of order 5

%e   0 1 2 3 4

%e   2 3 4 0 1

%e   4 0 1 2 3

%e   1 2 3 4 0

%e   3 4 0 1 2

%e has a(2)=5 diagonal transversals:

%e   0 . . . .   . 1 . . .   . . 2 . .   . . . 3 .   . . . . 4

%e   . . 4 . .   . . . 0 .   . . . . 1   2 . . . .   . 3 . . .

%e   . . . . 3   4 . . . .   . 0 . . .   . . 1 . .   . . . 2 .

%e   . 2 . . .   . . 3 . .   . . . 4 .   . . . . 0   1 . . . .

%e   . . . 1 .   . . . . 2   3 . . . .   . 4 . . .   . . 0 . .

%Y Cf. A006717, A123565, A287648, A338562, A341585, A342998.

%K nonn,more,hard

%O 0,3

%A _Eduard I. Vatutin_, Apr 02 2021