A354068 Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

1, 0, 0, 4, 5, 0, 8, 8, 14

Offset: 1

Author

Eduard I. Vatutin, May 16 2022

Keywords

• nonn
• more
• hard

Comments

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate.

a(10) <= 60, a(11) <= 279, a(12) <= 588, a(13) <= 9610.

Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= a(n) <= A360220(n) <= A287648(n). - Eduard I. Vatutin, Mar 03 2023

Examples

			One of the best orthogonal diagonal Latin squares of order n=9
  0 1 2 3 4 5 6 7 8
  1 2 3 8 6 4 7 0 5
  5 4 6 0 7 8 3 1 2
  7 3 1 5 2 6 0 8 4
  8 7 4 6 1 2 5 3 0
  3 0 5 4 8 7 1 2 6
  4 6 7 2 3 0 8 5 1
  6 5 8 1 0 3 2 4 7
  2 8 0 7 5 1 4 6 3
has orthogonal diagonal mate
  0 1 2 3 4 5 6 7 8
  2 3 8 7 5 6 4 1 0
  1 5 4 8 6 0 2 3 7
  8 7 0 6 1 3 5 4 2
  5 0 1 2 7 8 3 6 4
  4 6 7 0 3 2 8 5 1
  3 8 5 4 0 7 1 2 6
  7 4 6 5 2 1 0 8 3
  6 2 3 1 8 4 7 0 5
and 14 diagonal transversals, which is the minimal number, so a(9)=14.
		

Links

• Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
• E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
• Eduard I. Vatutin, Proving list (best known examples).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A287647, A287648, A345370, A349199, A360220.