A349199 -id:A349199 - cp's OEIS frontend

A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

1, 0, 0, 1, 2, 2, 14, 47, 182

Offset: 1

Eduard I. Vatutin, Jun 16 2021

a(n) <= A287648(n) - A287647(n) + 1.
a(n) <= A287764(n).
Conjecture: a(12) = A287648(12) - A287647(12) + 1. - Natalia Makarova, Oct 26 2021
a(10) >= 736, a(11) >= 1344, a(12) >= 17693, a(13) >= 18241, a(14) >= 294053, a(15) >= 1958394, a(16) >= 13715. - Eduard I. Vatutin, Oct 29 2021, updated Mar 02 2025

			For n=7 the number of diagonal transversals that a diagonal Latin square of order 7 may have is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, or 27. Since there are 14 distinct values, a(7)=14.

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, On the falsity of conjecture that spectra of diagonal transversals for diagonal Latin squares of order 12 is solid (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
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, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
E. I. Vatutin, Distributed diagonalization strategy for Latin squares, Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2023. pp. 309-311. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, 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)
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. A287647, A287648, A309344, A344105, A345760, A345761, A349199.

a(8) added by Eduard I. Vatutin, Jul 15 2021

a(9) added by Eduard I. Vatutin, Oct 20 2022

A350585 a(n) is the number of distinct numbers of transversals that an orthogonal diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 25, 295

Offset: 1

Eduard I. Vatutin, Mar 27 2022

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A344105(n).

a(10) >= 193, a(11) >= 3588, a(12) >= 10465. - updated by Eduard I. Vatutin, Jan 29 2023

			For n=8 the number of transversals that an orthogonal diagonal Latin square of order 8 may have is 16, 32, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 88, 96, 112, 128, 132, 144, 160, 168, 192, 224, 256, 320, or 384. Since there are 25 distinct values, a(8)=25.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Examples (1, 4, 5, 7, 8, 9, 10, 11, 12).
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, A344105, A349199.

A360220 Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 0, 27, 120, 333

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An orthogonal diagonal Latin square is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) >= 866, a(11) >= 4828, a(12) >= 30192, a(13) >= 131106, a(17) >= 204995269, a(19) >= 11254190082.
For most orders n, at least one diagonal Latin square with the maximal number of diagonal transversals has an orthogonal mate and A287648(n) = a(n). Known exceptions: n=6 and n=10. - Eduard I. Vatutin, Feb 17 2023
Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= A354068(n) <= a(n) <= A287648(n). - Eduard I. Vatutin, Mar 04 2023

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. A287647, A287648, A305570, A305571, A349199, A354068.

A354068 Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 4, 5, 0, 8, 8, 14

Offset: 1

Eduard I. Vatutin, May 16 2022

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate.
a(10) <= 60, a(11) <= 279, a(12) <= 588, a(13) <= 9610.
Every orthogonal diagonal Latin square is a diagonal Latin square, so A287647(n) <= a(n) <= A360220(n) <= A287648(n). - Eduard I. Vatutin, Mar 03 2023

			One of the best orthogonal diagonal Latin squares of order n=9
  0 1 2 3 4 5 6 7 8
  1 2 3 8 6 4 7 0 5
  5 4 6 0 7 8 3 1 2
  7 3 1 5 2 6 0 8 4
  8 7 4 6 1 2 5 3 0
  3 0 5 4 8 7 1 2 6
  4 6 7 2 3 0 8 5 1
  6 5 8 1 0 3 2 4 7
  2 8 0 7 5 1 4 6 3
has orthogonal diagonal mate
  0 1 2 3 4 5 6 7 8
  2 3 8 7 5 6 4 1 0
  1 5 4 8 6 0 2 3 7
  8 7 0 6 1 3 5 4 2
  5 0 1 2 7 8 3 6 4
  4 6 7 0 3 2 8 5 1
  3 8 5 4 0 7 1 2 6
  7 4 6 5 2 1 0 8 3
  6 2 3 1 8 4 7 0 5
and 14 diagonal transversals, which is the minimal number, so a(9)=14.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
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)
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A345370, A349199, A360220.

A354050 a(n) is the number of distinct numbers of intercalates that an orthogonal diagonal Latin square of order n can have.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 26, 55

Offset: 1

Eduard I. Vatutin, May 16 2022

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A345760(n).

a(10) >= 74, a(11) >= 76, a(12) >= 190. - updated by Eduard I. Vatutin, Mar 01 2025

			For n=8 the number of intercalates that an orthogonal diagonal Latin square of order 8 may have is 2, 4, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 36, 40, 44, 48, 52, 56, 64, 80, or 112. Since there are 26 distinct values, a(8)=26.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
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)
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.
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (4, 5, 7, 8, 9, 10, 11, 12).
Index entries for sequences related to Latin squares and rectangles.

Cf. A345760, A349199, A350585.

cp's OEIS Frontend

A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

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A350585 a(n) is the number of distinct numbers of transversals that an orthogonal diagonal Latin square of order n can have.

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A360220 Maximum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

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A354068 Minimum number of diagonal transversals in an orthogonal diagonal Latin square of order n.

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A354050 a(n) is the number of distinct numbers of intercalates that an orthogonal diagonal Latin square of order n can have.

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