A299783 Minimum size of a main class for diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 4, 32, 32, 96

Offset: 1

Author

Eduard I. Vatutin, Jan 21 2019

Keywords

• nonn
• more
• hard

Comments

a(9) <= 48; a(10) <= 1536, a(11) <= 1536, a(12) <= 46080, a(13) <= 7680. - Eduard I. Vatutin, Oct 05 2020, updated Apr 08 2025

Examples

			From _Eduard I. Vatutin_, Oct 05 2020: (Start)
The following DLS of order 9 has a main class with cardinality 48:
  0 1 2 3 4 5 6 7 8
  2 4 3 0 7 6 8 1 5
  6 2 8 5 3 4 7 0 1
  4 6 7 1 8 2 3 5 0
  1 5 4 7 6 0 2 8 3
  7 8 1 4 5 3 0 6 2
  3 7 0 2 1 8 5 4 6
  8 3 5 6 0 7 1 2 4
  5 0 6 8 2 1 4 3 7
The following DLS of order 10 has a main class with cardinality 7680:
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 3 6 5 9 7 8
  2 0 3 5 8 1 4 6 9 7
  4 6 9 7 1 8 2 0 3 5
  9 7 8 6 5 4 3 1 2 0
  3 4 7 8 0 9 1 2 5 6
  6 9 4 1 7 2 8 5 0 3
  7 8 5 0 6 3 9 4 1 2
  5 3 1 9 2 7 0 8 6 4
  8 5 6 2 9 0 7 3 4 1
(End)
		

Links

• E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 9 (in Russian).
• E. I. Vatutin, About the upper bound of the minimal size of main class for diagonal Latin squares of order 10 (in Russian).
• E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, and N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
• E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, and N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
• Eduard I. Vatutin, About the relationship between the minimal and maximal cardinality of main classes for diagonal Latin squares (in Russian).
• Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition — 2021, pp. 77-79. (in Russian)
• Eduard I. Vatutin, Proving list (best known examples).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A274171, A287764, A299784, A299785, A299787.

Formula

a(n) = A299785(n) / n!.

0 <= a(n) <= A299784(n). - Eduard I. Vatutin, Jun 08 2020

From Eduard I. Vatutin, added May 30 2021, updated Apr 08 2025: (Start)

a(n) = A299784(n) for 1 <= n <= 5.

a(6)*3 = A299784(6).

a(7)*6 = A299784(7).

a(8)*16 = A299784(8).

a(9)*32 <= A299784(9).

a(10)*10 <= A299784(10).

a(11)*10 <= A299784(11).

a(12)*4 <= A299784(12).

a(13)*24 <= A299784(13). (End)