A287761 Number of self-orthogonal diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 4, 0, 64, 1152, 224832, 234255360

Offset

1,4

Comments

A self-orthogonal diagonal Latin square is a diagonal Latin square orthogonal to its transpose.

A333367(n) <= a(n) <= A309598(n) <= A305570(n). - Eduard I. Vatutin, Apr 26 2020

Links

Table of n, a(n) for n=1..10.

E. I. Vatutin, About the number of SODLS of order 10, a(10) value is wrong (in Russian).

E. I. Vatutin, About the number of SODLS of order 10, corrected value a(10) (in Russian).

E. I. Vatutin, List of all main classes of self-orthogonal diagonal Latin squares of orders 1-10.

E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)

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.

Harry White, Self-orthogonal Diagonal Latin Squares. How many.

Index entries for sequences related to Latin squares and rectangles.

Formula

a(n) = A287762(n)/n!.

From Eduard I. Vatutin, Mar 14 2020: (Start)

a(i) != A329685(i)*A299784(i)/2 for i=1..9 due to the existence of doubly self-orthogonal diagonal Latin square (DSODLS) and/or generalized symmetries (automorphisms) for some SODLS.

a(10) = A329685(10)*A299784(10)/2 because no DSODLS exist for order n=10 and no SODLS of order n=10 have generalized symmetries (automorphisms). (End)

Example

0 1 2 3 4 5 6 7 8 9
5 2 0 9 7 8 1 4 6 3
9 5 7 1 8 6 4 3 0 2
7 8 6 4 9 2 5 1 3 0
8 9 5 0 3 4 2 6 7 1
3 6 9 5 2 1 7 0 4 8
4 3 1 7 6 0 8 2 9 5
6 7 8 2 5 3 0 9 1 4
2 0 4 6 1 9 3 8 5 7
1 4 3 8 0 7 9 5 2 6

Crossrefs

Cf. A160368, A287762, A329685.

Sequence in context: A012362 A012710 A333367 * A009512 A317411 A309598

Adjacent sequences: A287758 A287759 A287760 * A287762 A287763 A287764

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, May 31 2017

Extensions

a(10) from Eduard I. Vatutin, Mar 14 2020

a(10) corrected by Eduard I. Vatutin, Apr 24 2020

Status

approved