A339641 -id:A339641 - cp's OEIS frontend

A340186 Number of Brown's diagonal Latin squares of order 2n.

0, 48, 184320, 3948134400, 3470226200985600

Offset: 1

Eduard I. Vatutin, Dec 31 2020

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square. Diagonal Latin squares of this type have interesting properties, for example, a large number of transversals.

Plain symmetry diagonal Latin squares do not exist for odd orders, so a(2n+1)=0.

			The diagonal Latin square
.
   0 1 2 3 4 5 6 7 8 9
   1 2 3 4 0 9 5 6 7 8
   4 0 1 7 3 6 2 8 9 5
   8 7 6 5 9 0 4 3 2 1
   7 6 5 0 8 1 9 4 3 2
   9 8 7 6 5 4 3 2 1 0
   5 9 8 2 6 3 7 1 0 4
   3 5 0 8 7 2 1 9 4 6
   2 3 4 9 1 8 0 5 6 7
   6 4 9 1 2 7 8 0 5 3
.
is a Brown's square since it is horizontally symmetric (see A287649) and its rows form row-inverse pairs:
.
   0 1 2 3 4 5 6 7 8 9   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   1 2 3 4 0 9 5 6 7 8   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   4 0 1 7 3 6 2 8 9 5
   . . . . . . . . . .   8 7 6 5 9 0 4 3 2 1   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   9 8 7 6 5 4 3 2 1 0   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   5 9 8 2 6 3 7 1 0 4
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
.
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   7 6 5 0 8 1 9 4 3 2   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   3 5 0 8 7 2 1 9 4 6
   2 3 4 9 1 8 0 5 6 7   . . . . . . . . . .
   . . . . . . . . . .   6 4 9 1 2 7 8 0 5 3

J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, 1992, Vol. 139, pp. 43-49.

E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Eduard I. Vatutin, Enumeration of the Brown's diagonal Latin squares of orders 1-9 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A339305, A339641.

a(n) = A339305(n) * (2*n)!.

a(3) corrected by Eduard I. Vatutin and Oleg Zaikin, Jan 12 2025

a(5) added by Eduard I. Vatutin and Oleg S. Zaikin, Apr 02 2025

A339305 Number of Brown's diagonal Latin squares of order 2n with the first row in order.

Original entry on oeis.org

0, 2, 128, 97920, 956301312

Offset: 1

Eduard I. Vatutin, Dec 24 2020

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square. Diagonal Latin squares of this type have interesting properties, for example, a large number of transversals.

Plain symmetry diagonal Latin squares do not exist for odd orders, so a(2n+1)=0.

			The diagonal Latin square
.
   0 1 2 3 4 5 6 7 8 9
   1 2 3 4 0 9 5 6 7 8
   4 0 1 7 3 6 2 8 9 5
   8 7 6 5 9 0 4 3 2 1
   7 6 5 0 8 1 9 4 3 2
   9 8 7 6 5 4 3 2 1 0
   5 9 8 2 6 3 7 1 0 4
   3 5 0 8 7 2 1 9 4 6
   2 3 4 9 1 8 0 5 6 7
   6 4 9 1 2 7 8 0 5 3
.
is a Brown's square since it is horizontally symmetric (see A287649) and its rows form row-inverse pairs:
.
   0 1 2 3 4 5 6 7 8 9   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   1 2 3 4 0 9 5 6 7 8   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   4 0 1 7 3 6 2 8 9 5
   . . . . . . . . . .   8 7 6 5 9 0 4 3 2 1   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   9 8 7 6 5 4 3 2 1 0   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   5 9 8 2 6 3 7 1 0 4
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .   . . . . . . . . . .
.
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   7 6 5 0 8 1 9 4 3 2   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   . . . . . . . . . .
   . . . . . . . . . .   3 5 0 8 7 2 1 9 4 6
   2 3 4 9 1 8 0 5 6 7   . . . . . . . . . .
   . . . . . . . . . .   6 4 9 1 2 7 8 0 5 3

J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, 1992, Vol. 139, pp. 43-49.

E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Eduard I. Vatutin, Enumeration of the Brown's diagonal Latin squares of orders 1-9 (in Russian).
Eduard I. Vatutin, Clarification for Brown's diagonal Latin squares for orders 6 and 8 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287649, A339641, A340186, A379145.

a(n) = A340186(n) / (2*n)!. - Eduard I. Vatutin, Jan 08 2021

a(3) corrected by Eduard I. Vatutin and Oleg Zaikin, Dec 16 2024

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

A379145 Number of horizontal plane Brown's diagonal Latin squares of order 2n with the first row in order.

Original entry on oeis.org

0, 2, 64, 49152, 478150656

Offset: 1

Eduard I. Vatutin, Dec 16 2024

A Brown's diagonal Latin square is a horizontally symmetric row-inverse (horizontal plane Brown's diagonal Latin square) or vertically symmetric column-inverse diagonal Latin square (vertical plane Brown's diagonal Latin square). Diagonal Latin squares of this type have interesting properties, for example, a large number of transversals.
Also number of vertical plane Brown's diagonal Latin squares of order 2n with the first row in order.
Plain symmetry diagonal Latin squares do not exist for odd orders.

E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Eduard I. Vatutin, Enumeration of the Brown's diagonal Latin squares of orders 1-9 (in Russian).
Eduard I. Vatutin, Clarification for Brown's diagonal Latin squares for orders 6 and 8 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287649, A339305, A339641, A340186, A381626.

a(n) = A381626(n) / (2n)!.

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

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

A381971 Maximum number of diagonal transversals in a Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 4, 6, 120, 890

Offset: 1

Eduard I. Vatutin, Mar 11 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 are not exist for odd orders.
a(6)>=28496, a(7)>=490218, a(8)>=32172800.

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. A287648, A339641.

A381626 Number of horizontal plane Brown's diagonal Latin squares of order 2n.

Original entry on oeis.org

0, 48, 92160, 1981808640, 1735113100492800

Offset: 1

Eduard I. Vatutin, Mar 02 2025

 Brown's diagonal Latin square is a horizontally symmetric row-inverse (horizontal plane Brown's diagonal Latin square) or vertically symmetric column-inverse diagonal Latin square (vertical plane Brown's diagonal Latin square). Diagonal Latin squares of this type have interesting properties, for example, a large number of transversals.
Also number of vertical plane Brown's diagonal Latin squares of order 2n with the first row in order.
Plain symmetry diagonal Latin squares do not exist for odd orders.

E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022), Pereslavl-Zalessky, 2023, pp. 9-18 (in Russian).
Eduard I. Vatutin, Enumeration of the Brown's diagonal Latin squares of orders 1-9 (in Russian).
Eduard I. Vatutin, Clarification for Brown's diagonal Latin squares for orders 6 and 8 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A292516, A339641, A340186, A379145.

a(n) = A379145(n) * (2n)!.

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

A382024 Maximum number of transversals in a Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 8, 32, 384, 5504

Offset: 1

Eduard I. Vatutin, Mar 12 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)>=198144, a(7)>=3477504, a(8)>=244744192.

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. A287644, A339641.

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.

A382272 Maximum number of orthogonal diagonal Latin squares with the first row in ascending order that can be orthogonal to a given Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 1, 0, 824, 8

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)>=1764493860.

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. A287695, A339641.

A382505 a(n) is the number of distinct numbers of diagonal transversals in Brown's diagonal Latin squares of order 2n.

Original entry on oeis.org

0, 1, 2, 20, 349

Offset: 1

Eduard I. Vatutin, Mar 29 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)>=1785, a(7)>=60341, a(8)>=4151.

			For n=4 the number of transversals that a diagonal Latin square of order 8 may have is 0, 8, 12, 16, 18, 20, 24, 26, 28, 32, 36, 40, 44, 48, 52, 56, 64, 88, 96, or 120. Since there are 20 distinct values, a(4)=20.

Eduard I. Vatutin, About the spectra of numerical characteristics of Brown's diagonal Latin squares (in Russian).
Eduard I. Vatutin, Proving lists (4, 6, 8, 10).
Eduard I. Vatutin, Graphical representation of the spectra.
Index entries for sequences related to Latin squares and rectangles.

Cf. A339641, A344105, A381971.

cp's OEIS Frontend

A340186 Number of Brown's diagonal Latin squares of order 2n.

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A339305 Number of Brown's diagonal Latin squares of order 2n with the first row in order.

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A379145 Number of horizontal plane Brown's diagonal Latin squares of order 2n with the first row in order.

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

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A381971 Maximum number of diagonal transversals in a Brown's diagonal Latin square of order 2n.

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A381626 Number of horizontal plane Brown's diagonal Latin squares of order 2n.

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A382024 Maximum number of transversals 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.

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A382272 Maximum number of orthogonal diagonal Latin squares with the first row in ascending order that can be orthogonal to a given Brown's diagonal Latin square of order 2n.

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A382505 a(n) is the number of distinct numbers of diagonal transversals in Brown's diagonal Latin squares of order 2n.

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