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

1,4

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

Links

Table of n, a(n) for n=1..9.

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.

Example

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.

Crossrefs

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

Sequence in context: A249860 A360962 A320162 * A200013 A178219 A322232

Adjacent sequences: A354065 A354066 A354067 * A354069 A354070 A354071

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, May 16 2022

Status

approved