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

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

Offset: 1

Author

Eduard I. Vatutin, Jan 21 2019

Keywords

• nonn
• more
• hard

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

Examples

			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)
		

Links

• 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.

Crossrefs

Cf. A287764, A299783, A299784, A299787.

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)