A293777 Number of centrally symmetric diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 8, 0, 2816, 135168, 327254016

Offset: 1

Author

Eduard I. Vatutin, Oct 16 2017

Keywords

• nonn
• more
• hard

Comments

A centrally symmetric diagonal Latin square is a square with one-to-one correspondence between elements within all pairs a[i][j] and a[n-1-i][n-1-j] (with rows and columns numbered from 0 to n-1).

a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist). It seems that a(n)=0 for n == 2 (mod 4).

Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(4n) >= A287650(n). - Eduard I. Vatutin, May 03 2021

Examples

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

Links

• Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
• Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).
• E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Properties of central symmetry for diagonal Latin squares, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
• E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
• 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)
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

Cf. A287649, A287650, A293778, A340545.

Formula

a(n) = A293778(n) / n!.