A299784 Maximum size of a main class for diagonal Latin squares of order n with the first row in ascending order.

1, 0, 0, 2, 4, 96, 192, 1536, 1536, 15360, 15360, 184320, 184320, 2580480, 2580480

Offset: 1

Author

Eduard I. Vatutin, Jan 21 2019

Keywords

• nonn
• more
• hard
• changed

Comments

a(n) <= 2^m * m! * 4, where m = floor(n/2).

It seems that a(n) = 2^m * m! * 4 for all n > 6. - Eduard I. Vatutin, Jun 08 2020

0 <= A299783(n) <= a(n). - Eduard I. Vatutin, Jun 08 2020

Examples

			From _Eduard I. Vatutin_, May 30 2021: (Start)
The following DLS of order 9 has a main class with cardinality 1536:
  0 1 2 3 4 5 6 7 8
  1 2 0 4 8 6 5 3 7
  7 4 5 8 0 3 2 6 1
  5 8 7 6 1 0 3 2 4
  8 0 3 2 7 1 4 5 6
  3 7 8 5 6 4 1 0 2
  6 3 1 7 5 2 8 4 0
  2 6 4 0 3 8 7 1 5
  4 5 6 1 2 7 0 8 3
The following DLS of order 10 has a main class with cardinality 15360:
  0 1 2 3 4 5 6 7 8 9
  1 2 0 4 5 3 9 8 6 7
  3 5 6 1 8 7 4 0 9 2
  9 4 7 8 3 2 1 6 0 5
  2 7 3 0 9 8 5 1 4 6
  6 8 5 9 2 4 7 3 1 0
  4 6 9 7 0 1 3 2 5 8
  7 0 4 6 1 9 8 5 2 3
  8 3 1 5 6 0 2 9 7 4
  5 9 8 2 7 6 0 4 3 1
(End)
		

Links

• 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.
• E. I. Vatutin, Discussion about properties of diagonal Latin squares (in Russian).
• E. I. Vatutin, About the maximal size of main class for diagonal Latin squares of orders 11-15 (in Russian).
• Eduard I. Vatutin, Estimating the maximal size of main class for diagonal Latin squares of orders 9-15, Medical-Ecological and Information Technologies - 2020, Part 2, 2020, pp. 57-62. (in Russian)
• 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.

Formula

a(n) = A299787(n) / n!.

From Eduard I. Vatutin, May 30 2021: (Start)

A299783(n) = a(n) for 1 <= n <= 5.

A299783(6)*3 = a(6).

A299783(7)*6 = a(7).

A299783(8)*16 = a(8).

A299783(9)*32 <= a(9).

A299783(10)*10 <= a(10).

A299783(11)*10 <= a(11).

A299783(12)*4 <= a(12).

A299783(13)*24 <= a(13). (End)

Extensions

a(9)-a(10) from Eduard I. Vatutin, Mar 15 2020

a(11)-a(15) from Eduard I. Vatutin, Jun 08 2020