A307167 Maximum number of loops in a diagonal Latin square of order n.

1, 0, 0, 12, 14, 27, 53, 112

Offset

1,4

Comments

A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}.

For diagonal Latin squares of order 4 all loops are intercalates. - Eduard I. Vatutin, Oct 05 2020

Links

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

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

Eduard I. Vatutin, About the minimum and maximum number of loops in a diagonal Latin squares of order 8 (in Russian).

Eduard I. Vatutin, Proving list (best known examples).

Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.

Index entries for sequences related to Latin squares and rectangles.

Example

For example, the square
  2 4 3 5 0 1
  1 0 4 3 2 5
  0 2 5 4 1 3
  5 3 0 1 4 2
  4 5 1 2 3 0
  3 1 2 0 5 4
has a loop
  2 4 . . . .
  . . . . . .
  . 2 . 4 . .
  . . . . . .
  4 . . 2 . .
  . . . . . .
consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6.
The total number of loops for this square is 21.

Crossrefs

Cf. A274806, A307166, A307163, A307164.

Sequence in context: A175886 A181451 A022326 * A238228 A214504 A140810

Adjacent sequences: A307164 A307165 A307166 * A307168 A307169 A307170

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Mar 27 2019

Extensions

a(8) added by Eduard I. Vatutin, Oct 05 2020

Status

approved