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 of length < 2*n.
LINKS
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimum and maximum number of partial loops in a diagonal Latin squares of order 8 (in Russian).
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.
Eduard I. Vatutin, Proving list (best known examples).
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
KEYWORD
nonn,more,hard
AUTHOR
Eduard I. Vatutin, Mar 27 2019
EXTENSIONS
a(8) added by Eduard I. Vatutin, Oct 06 2020
STATUS
approved