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

Author

Eduard I. Vatutin, May 31 2017

Keywords

• nonn
• more
• hard

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

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A160368, A287762, A329685.

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)

Extensions

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

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