A340545 Number of main classes of centrally symmetric diagonal Latin squares of order n.

1, 0, 0, 1, 2, 0, 32, 301, 430090

Offset

1,5

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 numbering of rows and columns from 0 to n-1).

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

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

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, 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, and 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)

E. I. Vatutin, About the number of main classes of centrally symmetric diagonal Latin squares of orders 1-9 (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.

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

Cf. A293777, A293778, A287764, A340550.

Sequence in context: A156452 A156473 A156504 * A009665 A053552 A009548

Adjacent sequences: A340542 A340543 A340544 * A340546 A340547 A340548

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Jan 11 2021

Status

approved