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

1,4

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

Links

Table of n, a(n) for n=1..8.

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.

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)

Example

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)

Crossrefs

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

Sequence in context: A083205 A019542 A319223 * A293760 A101575 A197099

Adjacent sequences: A299780 A299781 A299782 * A299784 A299785 A299786

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Jan 21 2019

Status

approved