A293778 - OEIS

A293778

Number of centrally symmetric diagonal Latin squares of order n.

1, 0, 0, 48, 960, 0, 14192640, 5449973760, 118753937326080

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OFFSET

1,4

COMMENTS

Centrally symmetric diagon 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] (numbering of rows and columns from 0 to n-1).

It seems that a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist) and 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 A292517(n) <= a(4n). - Eduard I. Vatutin, May 26 2021

LINKS

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

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (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)

Eduard I. Vatutin, On the interconnection between double and central symmetries in diagonal Latin squares (in Russian).

Index entries for sequences related to Latin squares and rectangles.

FORMULA

a(n) = A293777(n) * n!.

EXAMPLE

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