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
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.
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
KEYWORD
nonn,more,hard
AUTHOR
Eduard I. Vatutin, Mar 27 2019
EXTENSIONS
a(8) added by Eduard I. Vatutin, Oct 05 2020
STATUS
approved