A299787 Maximum size of a main class for diagonal Latin squares of order n.

1, 0, 0, 48, 480, 69120, 967680, 61931520, 557383680, 55738368000, 613122048000, 88289574912000, 1147764473856000, 224961836875776000, 3374427553136640000

Offset: 1

Author

Eduard I. Vatutin, Jan 21 2019

Keywords

• nonn
• more

Comments

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

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

0 <= A299785(n) <= a(n). - Eduard I. Vatutin, Jul 06 2020

Examples

			From _Eduard I. Vatutin_, May 31 2021: (Start)
The following DLS of order 9 has a main class with cardinality 1536*9! = 557383680:
  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*10! = 55738368000:
  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. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
• E. I. Vatutin, About the maximal size of main classes of diagonal Latin squares of orders 9 and 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.
• E. I. Vatutin, About the maximal size of main class for diagonal Latin squares of orders 11-15 (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, 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, Proving list (best known examples).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A287764, A299783, A299784, A299785.

Formula

a(n) = A299784(n) * n!.

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

a(n) = A299785(n) for 1 <= n <= 5.

a(6) = A299785(6)*3.

a(7) = A299785(7)*6.

a(8) = A299785(8)*16.

a(9) = A299785(9)*32.

a(10) = A299785(10)*2.

a(11) = A299785(11)*10.

a(12) = A299785(12)*4.

a(13) = A299785(13)*24. (End)

Extensions

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

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