A307839 -id:A307839 - cp's OEIS frontend

A307841 Minimum number of nontrivial Latin subrectangles in a diagonal Latin square of order n.

0, 0, 0, 12, 0, 51, 0, 36

Offset: 1

Eduard I. Vatutin, May 01 2019

A Latin subrectangle is an m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.

A nontrivial Latin subrectangle is an m X k Latin rectangle of a Latin square of order n, 1 < m < n, 1 < k < n.

			For example, the square
  0 1 2 3 4 5 6
  4 2 6 5 0 1 3
  3 6 1 0 5 2 4
  6 3 5 4 1 0 2
  1 5 3 2 6 4 0
  5 0 4 6 2 3 1
  2 4 0 1 3 6 5
has a nontrivial Latin subrectangle
  . . . . . . .
  . . 6 5 0 1 3
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . 0 1 3 6 5
The total number of Latin subrectangles for this square is 2119 and the number of nontrivial Latin subrectangles is only 151.

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 nontrivial Latin subrectangles 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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A307839, A307842.

a(8) added by Eduard I. Vatutin, Oct 06 2020

A307840 Maximum number of Latin subrectangles in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 137, 348, 884, 2119, 5433

Offset: 1

Eduard I. Vatutin, May 01 2019

An Latin subrectangle is a m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.

			For example, the square
  0 1 2 3 4 5 6
  4 2 6 5 0 1 3
  3 6 1 0 5 2 4
  6 3 5 4 1 0 2
  1 5 3 2 6 4 0
  5 0 4 6 2 3 1
  2 4 0 1 3 6 5
has a Latin subrectangle
  . . . . . . .
  . . 6 5 0 1 3
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . 0 1 3 6 5
The total number of Latin subrectangles for this square is 2119.

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 Latin subrectangles 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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307163, A307164, A307839, A307841, A307842.

a(8) added by Eduard I. Vatutin, Oct 06 2020

cp's OEIS Frontend

A307841 Minimum number of nontrivial Latin subrectangles in a diagonal Latin square of order n.

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