A342998 Minimum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

1, 0, 5, 27, 0, 4523, 128818, 0, 204330233, 11232045257

Offset: 0

Author

Eduard I. Vatutin, Apr 02 2021

Keywords

• nonn
• more
• hard

Comments

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).

Cyclic diagonal Latin squares do not exist for even orders.

a(n) <= A342997(n).

All cyclic diagonal Latin squares are diagonal Latin squares, so A287647(n) <= a((n-1)/2).

Examples

			For n=2 one of best cyclic diagonal Latin squares of order 5
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
has a(2)=5 diagonal transversals:
  0 . . . .   . 1 . . .   . . 2 . .   . . . 3 .   . . . . 4
  . . 4 . .   . . . 0 .   . . . . 1   2 . . . .   . 3 . . .
  . . . . 3   4 . . . .   . 0 . . .   . . 1 . .   . . . 2 .
  . 2 . . .   . . 3 . .   . . . 4 .   . . . . 0   1 . . . .
  . . . 1 .   . . . . 2   3 . . . .   . 4 . . .   . . 0 . .
		

Links

• Eduard I. Vatutin, Enumerating the diagonal transversals for cyclic diagonal Latin squares of orders 1-19 (in Russian).
• Eduard I. Vatutin, Proving list (best known examples).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A006717, A123565, A287647, A287648, A338562, A341585, A342997.