A307170 Minimum number of partial loops in a diagonal Latin square of order n.

0, 0, 0, 12, 0, 21, 0, 24

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}. A partial loop is a loop with length < 2*n.

From Eduard I. Vatutin, Oct 20 2020: (Start)

Every intercalate is a partial loop and every partial loop is a loop, so 0 <= A307163(n) <= a(n) <= A307166(n).

0 <= a(n) <= A307171(n).

(End)

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 partial loops in a diagonal Latin squares of order 8 (in Russian).

Eduard I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (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 < 12.
The total number of loops for this square is 21, all of which are partial.

Crossrefs

Cf. A274806, A307166, A307167, A307163, A307171.

Sequence in context: A254526 A156390 A059680 * A332053 A225951 A333577

Adjacent sequences: A307167 A307168 A307169 * A307171 A307172 A307173

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Mar 27 2019

Extensions

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

Status

approved