A307163 -id:A307163 - cp's OEIS frontend

A307164 Maximum number of intercalates in a diagonal Latin square of order n.

0, 0, 0, 12, 4, 9, 30, 112, 72

Offset: 1

Eduard I. Vatutin, Mar 27 2019

An intercalate is a 2 X 2 subsquare of a Latin square.
0 <= A307163(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020
a(10) >= 109, a(11) >= 172, a(12) >= 324, a(13) >= 180, a(14) >= 391, a(15) >= 630, a(16) >= 960, a(17) >= 736, a(18) >= 547, a(19) >= 457, a(20) >= 1100, a(21) >= 785, a(22) >= 887, a(23) >= 899, a(24) >= 1680, a(25) >= 1700, a(26) >= 1299, a(27) >= 1372, a(28) >= 2892. - Eduard I. Vatutin, May 31 2021, updated Mar 02 2025
If, in theory, all unordered pairs of rows and columns form intercalate in their intersection, total number of intercalates will be (n*(n-1))^2, so a(n) <= (n*(n-1))^2, a(n) is asymptotically less than O(n^4). In practice a(n) << (n*(n-1))^2. - Eduard I. Vatutin, Mar 05 2025

			From _Eduard I. Vatutin_, May 31 2021: (Start)
One of the best known diagonal Latin squares of order n=5
  0 1 2 3 4
  4 2 0 1 3
  1 4 3 2 0
  3 0 1 4 2
  2 3 4 0 1
has 4 intercalates:
  . . 2 3 .   . . . . .   . . . . .   . . . . .
  . . . . .   . . 0 . 3   . . . . .   . . . . .
  . . 3 2 .   . . 3 . 0   1 . 3 . .   . 4 3 . .
  . . . . .   . . . . .   3 . 1 . .   . . . . .
  . . . . .   . . . . .   . . . . .   . 3 4 . .
so a(5)=4. (End)

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, About the maximum number of intercalates in a diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14 (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, Cham (2020), 127-146.
Eduard I. Vatutin, Natalia N. Nikitina, and Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Alexandr M. Albertyan, and Ilya I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17 (in Russian).
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307163, A345760.

a(9) added by Eduard I. Vatutin, Sep 21 2020

A345760 a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 21, 61, 64

Offset: 1

Eduard I. Vatutin, Jun 26 2021

a(n) <= A307164(n) - A307163(n) + 1.
a(n) <= A287764(n).
a(10) >= 98, a(11) >= 145, a(12) >= 259, a(13) >= 200, a(14) >= 362, a(15) >= 536, a(16) >= 792, a(17) >= 685, a(18) >= 535, a(19) >= 447, a(20) >= 1011, a(21) >= 747, a(22) >= 872, a(23) >= 885, a(24) >= 1610, a(25) >= 1677, a(26) >= 1266, a(27) >= 1337, a(28) >= 2795. - Eduard I. Vatutin, Oct 02 2021, updated Mar 02 2025

			For n=7 the number of intercalates that a diagonal Latin square of order 7 may have is 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26, or 30. Since there are 21 distinct values, a(7)=21.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A307163, A307164, A309344, A344105, A345370, A345761.

a(9) added by Eduard I. Vatutin, Oct 22 2022

A307166 Minimum number of loops in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 12, 10, 27, 21, 40

Offset: 1

Eduard I. Vatutin, Mar 27 2019

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
From Eduard I. Vatutin, Oct 26 2020: (Start)
Every intercalate is a partial loop and every partial loop is a loop, so 0 <= A307163(n) <= A307170(n) <= a(n).
0 <= a(n) <= A307167(n).
(End)

			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.

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 loops in a diagonal Latin squares of order 8 (in Russian).
E. 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 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, A307167, A307170.

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

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

Original entry on oeis.org

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

Offset: 1

Eduard I. Vatutin, Mar 27 2019

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

			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.

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.

Cf. A274806, A307166, A307163, A307164.

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

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

Original entry on oeis.org

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

Offset: 1

Eduard I. Vatutin, Mar 27 2019

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)

			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.

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.

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

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

A307839 Minimum number of Latin subrectangles in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 137, 336, 884, 1968, 4545

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, A307840, A307841, 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

A360221 Minimum number of intercalates in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 0, 0, 0, 2, 0

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An intercalate is a 2 X 2 subsquare of a Latin square.
An orthogonal diagonal Latin squares is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) <= 1, a(11) = 0, a(12) <= 4, a(13) = 0. - Eduard I. Vatutin, added Jan 30 2023, updated Sep 24 2024

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A305570, A305571, A307163, A354050, A360223.

A379665 Minimum number of intercalates in a Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 0, 12, 9, 16, 25

Offset: 0

Eduard I. Vatutin, Dec 29 2024

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square (see A339641).
Plain symmetry diagonal Latin squares do not exist for odd orders.
a(6)<=36, a(7)<=49, a(8)<=64, a(9)<=81, a(10)<=100, a(11)<=121, a(12)<=144, a(13)<=201, a(14)<=252. - Updated by Eduard I. Vatutin, Mar 01 2025
Hypothesis: minimum number of intercalates in Brown's diagonal Latin squares of order N=2n is equal to (N/2)^2 for N>4 (proved for N=6 and N=8 using Brute Force and for 10<=N<=24 using heuristic methods).

Eduard I. Vatutin, On the number of intercalates in diagonal Latin squares of special type, Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2024. pp. 284-286. (in Russian)
Eduard I. Vatutin, About the minimum number of intercalates in Brown's diagonal Latin squares of orders N<25 (in Russian).
Eduard I. Vatutin, Proving list (best known examples)
Index entries for sequences related to Latin squares and rectangles.

Cf. A307163, A339641.

a(5)=25 added by Oleg S. Zaikin and Eduard I. Vatutin, Apr 08 2025

A382270 Maximum number of intercalates in a Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 12, 9, 112, 57

Offset: 1

Eduard I. Vatutin, Mar 20 2025

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square (see A339641).
Brown's diagonal Latin squares are special case of plain symmetry diagonal Latin squares that do not exist for odd orders.
a(6)>=252, a(7)>=385, a(8)>=960, a(9)>=329, a(10)>=356, a(11)>=497, a(12)>=1008, a(13)>=497, a(14)>=524.

Eduard I. Vatutin, About the spectra of numerical characteristics of Brown's diagonal Latin squares (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307163, A307164, A339641, A345760, A368182, A379665.

cp's OEIS Frontend

A307164 Maximum number of intercalates in a diagonal Latin square of order n.

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A345760 a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.

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A307166 Minimum number of loops in a diagonal Latin square of order n.

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

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

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A307839 Minimum number of Latin subrectangles in a diagonal Latin square of order n.

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

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A360221 Minimum number of intercalates in an orthogonal diagonal Latin square of order n.

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A379665 Minimum number of intercalates in a Brown's diagonal Latin square of order 2n.

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A382270 Maximum number of intercalates in a Brown's diagonal Latin square of order 2n.