A299785 Minimum size of a main class for diagonal Latin squares of order n.

1, 0, 0, 48, 480, 23040, 161280, 3870720

Offset

1,4

Comments

0 <= a(n) <= A299787(n). - Eduard I. Vatutin, Jun 08 2020

a(9) <= 17418240; a(10) <= 27869184000. - Eduard I. Vatutin, Oct 05 2020

a(11) <= 61312204800, a(12) <= 22072393728000, a(13) <= 47823519744000. - Eduard I. Vatutin, May 31 2021

Links

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

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

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, 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, 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, Proving list (best known examples).

Index entries for sequences related to Latin squares and rectangles.

Formula

a(n) = A299783(n) * n!.

From Eduard I. Vatutin, May 31 2021: (Start)

a(n) = A299787(n) for 1 <= n <= 5.

a(6) = A299787(6)/3.

a(7) = A299787(7)/6.

a(8) = A299787(8)/16.

a(9) = A299787(9)/32.

a(10) = A299787(10)/2.

a(11) = A299787(11)/10.

a(12) = A299787(12)/4.

a(13) = A299787(13)/24. (End)

Example

From Eduard I. Vatutin, Oct 05 2020: (Start)
The following DLS of order 9 has a main class with cardinality 48*9! = 17418240:
  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*10! = 27869184000:
  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. A287764, A299783, A299784, A299787.

Sequence in context: A309599 A305571 A229505 * A299787 A168351 A198398

Adjacent sequences: A299782 A299783 A299784 * A299786 A299787 A299788

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Jan 21 2019

Status

approved