

A340545


Number of main classes of centrally symmetric diagonal Latin squares of order n.


3




OFFSET

1,5


COMMENTS

A centrally symmetric diagonal Latin square is a square with onetoone correspondence between elements within all pairs a[i, j] and a[n1i, n1j] (with numbering of rows and columns from 0 to n1).
It seems that a(n)=0 for n==2 (mod 4).
Centrally symmetric Latin squares are Latin squares, so a(n) <= A287764(n).
The canonical form (CF) of a square is the lexicographically minimal item within the corresponding main class of diagonal Latin square.
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A340550(n) <= a(n).  Eduard I. Vatutin, May 28 2021


LINKS

E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, Properties of central symmetry for diagonal Latin squares, Highperformance computing systems and technologies, No. 1 (8), 2018, pp. 7478. (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, Central Symmetry Properties for Diagonal Latin Squares, Problems of Information Technology, No. 2, 2019, pp. 38. 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). PereslavlZalessky, 2023. pp. 918. (in Russian)


EXAMPLE

For n=4 there is a single CF:
0 1 2 3
2 3 0 1
3 2 1 0
1 0 3 2
so a(4)=1.
For n=5 there are two different CFs:
0 1 2 3 4 0 1 2 3 4
2 3 4 0 1 1 3 4 2 0
4 0 1 2 3 4 2 1 0 3
1 2 3 4 0 2 0 3 4 1
3 4 0 1 2 3 4 0 1 2
so a(5)=2.
Example of a centrally symmetric diagonal Latin square of order n=9:
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


CROSSREFS



KEYWORD

nonn,more,hard


AUTHOR



STATUS

approved



