A309598 Number of extended self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 4, 0, 256, 4608

Offset

1,4

Comments

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

A333367(n) <= A287761(n) <= a(n) <= A305570(n). - Eduard I. Vatutin, Jun 07 2020

a(10) >= 510566400. - Eduard I. Vatutin, Jul 10 2020

Links

Table of n, a(n) for n=1..8.

E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian).

E. I. Vatutin, About the lower bound of number of ESODLS of order 10 (in Russian).

E. I. Vatutin, List of all main classes of extended self-orthogonal diagonal Latin squares of orders 1-8.

E. I. Vatutin and A. D. Belyshev, About the number of self-orthogonal (SODLS) and doubly self-orthogonal diagonal Latin squares (DSODLS) of orders 1-10. High-performance computing systems and technologies. Vol. 4. No. 1. 2020. pp. 58-63. (in Russian)

E. Vatutin and A. Belyshev, Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality, Communications in Computer and Information Science, Vol. 1331, Springer, 2020, pp. 586-597.

Index entries for sequences related to Latin squares and rectangles

Formula

From Eduard I. Vatutin, Feb 25 2020: (Start)

a(n) = A287761(n) for 1 <= n <= 6.

a(n) = 4*A287761(n) for 7 <= n <= 8. (End)

a(10) = A309210(10)*A299784(10) because no DSODLS exist for order n=10 and no ESODLS of order n=10 have generalized M-symmetries (automorphisms). - Eduard I. Vatutin, Jul 10 2020

Example

The diagonal Latin square
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 5 7 9 8 6 3
  5 0 1 6 3 9 8 2 4 7
  9 3 5 8 2 1 7 4 0 6
  4 6 3 5 7 8 0 9 2 1
  8 4 6 9 1 3 2 5 7 0
  7 8 9 0 6 4 5 1 3 2
  2 9 4 7 8 0 3 6 1 5
  6 5 7 1 0 2 4 3 9 8
  3 7 8 2 9 6 1 0 5 4
has orthogonal diagonal Latin square
  0 1 2 3 4 5 6 7 8 9
  3 5 9 8 6 2 0 1 4 7
  4 3 8 7 2 1 9 0 5 6
  6 9 3 4 8 0 1 2 7 5
  7 2 0 1 9 3 5 8 6 4
  2 0 1 5 7 6 4 9 3 8
  8 6 4 2 0 9 7 5 1 3
  1 7 6 0 5 4 8 3 9 2
  9 8 5 6 1 7 3 4 2 0
  5 4 7 9 3 8 2 6 0 1
from the same main class.

Crossrefs

Cf. A287761, A305570, A333367.

Sequence in context: A287761 A009512 A317411 * A305570 A287651 A365951

Adjacent sequences: A309595 A309596 A309597 * A309599 A309600 A309601

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Aug 09 2019

Status

approved