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

1,4

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

Links

Table of n, a(n) for n=1..15.

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.

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)

Example

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)

Crossrefs

Cf. A287764, A299783, A299784, A299785.

Sequence in context: A305571 A229505 A299785 * A168351 A198398 A211149

Adjacent sequences: A299784 A299785 A299786 * A299788 A299789 A299790

Keyword

nonn,more

Author

Eduard I. Vatutin, Jan 21 2019

Extensions

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

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

Status

approved