A348212 -id:A348212 - cp's OEIS frontend

A287644 Maximum number of transversals in a diagonal Latin square of order n.

1, 0, 0, 8, 15, 32, 133, 384, 2241

Offset: 1

Eduard I. Vatutin, May 29 2017

Same as the maximum number of transversals in a Latin square of order n except n = 3.
a(10) >= 5504 from Parker and Brown.
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= A357514(n) <= a(n) <= A090741(n). - Eduard I. Vatutin, added Sep 20 2020, updated Mar 03 2023
a(11) >= 37851, a(12) >= 198144, a(13) >= 1030367, a(14) >= 3477504, a(15) >= 36362925, a(16) >= 244744192, a(17) >= 1606008513, a(19) >= 87656896891, a(23) >= 452794797220965, a(25) >= 41609568918940625. - Eduard I. Vatutin, Mar 08 2020, updated Mar 10 2022
Also a(n) is the maximum number of transversals in an orthogonal diagonal Latin square of order n for all orders except n=6 where orthogonal diagonal Latin squares don't exist. - Eduard I. Vatutin, Jan 23 2022
All cyclic diagonal Latin squares are diagonal Latin squares, so A348212((n-1)/2) <= a(n) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021

J. W. Brown et al., Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, volume 139 (1992), pp. 43-49.
E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), pp. 73-81.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru.
E. I. Vatutin, About the minimal and maximal number of transversals in a diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, Enumerating the Main Classes of Cyclic and Pandiagonal Latin Squares, Recognition — 2021, pp. 77-79. (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, Cham (2020), 127-146.
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, Diagonalization and Canonization of Latin Squares, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian).
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, and V. S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017, pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares. Open Engineering. Vol. 7. Iss. 1. 2017, pp. 453-460. DOI: 10.1515/eng-2017-0052
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1, pp. 12-21 (in Russian).
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).
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, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
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, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A090741, A287645, A287647, A287648, A344105, A350585, A357514.

a(8) added by Eduard I. Vatutin, Oct 29 2017

a(9) added by Eduard I. Vatutin, Sep 20 2020

A287645 Minimum number of transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 3, 32, 7, 8, 68

Offset: 1

Eduard I. Vatutin, May 29 2017

From Eduard I. Vatutin, Sep 20 2020: (Start)
Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A287644(n) <= A090741(n).
A lower bound for odd n is A091323((n-1)/2) <= a(n). (End)
By definition, the main diagonal and antidiagonal of a diagonal Latin square are transversals, so a(n)>=2 for all n>=4 (the two diagonals are the same in the order 1 square and there are no diagonal Latin squares of orders 2 or 3). - Eduard I. Vatutin, Jun 13 2021
All cyclic diagonal Latin squares are diagonal Latin squares, so a(n) <= A348212((n-1)/2) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021
a(10) <= 128, a(11) <= 814, a(12) <= 448, a(13) <= 43093, a(14) <= 25720, a(15) <= 215721, a(16) <= 7465984. - Eduard I. Vatutin, Mar 11 2021, updated Feb 12 2025

			From _Eduard I. Vatutin_, Apr 24 2021: (Start)
For example, diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals (see A287648)
  0 . . .   . 1 . .   . . 2 .   . . . 3
  . . 1 .   . . . 0   3 . . .   . 2 . .
  . . . 2   . . 3 .   . 0 . .   1 . . .
  . 3 . .   2 . . .   . . . 1   . . 0 .
and 4 not diagonal transversals
  0 . . .   . 1 . .   . . 2 .   . . . 3
  . 2 . .   3 . . .   . . . 0   . . 1 .
  . . 3 .   . . . 2   1 . . .   . 0 . .
  . . . 1   . . 0 .   . 3 . .   2 . . .
total 8 transversals. (End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimal and maximal number of transversals in diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, Best examples presently known.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares. Open Engineering. Vol. 7. Iss. 1. 2017. pp. 453-460. DOI: 10.1515/eng-2017-0052
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev, and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1. pp. 12-21 (in Russian).
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).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
E. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.
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, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, and 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.
Index entries for sequences related to Latin squares and rectangles.

Cf. A091323, A287644, A287647, A287648, A344105.

a(8) added by Eduard I. Vatutin, Oct 29 2017

a(9) added by Eduard I. Vatutin, Sep 20 2020

A366332 Minimum number of diagonal transversals in a semicyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 0, 4523, 127339, 0, 204330233, 11232045257, 0

Offset: 0

Eduard I. Vatutin, Oct 07 2023

A horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). Similarly, a vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i).

			Example of horizontally semicyclic diagonal Latin square of order 13:
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=1)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)

Eduard I. Vatutin, About the spectra of numerical characteristics of different types of cyclic diagonal Latin squsres (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 17 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 19 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A071607, A342990, A342997, A342998, A348212, A366331.

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A287644 Maximum number of transversals in a diagonal Latin square of order n.

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A287645 Minimum number of transversals in a diagonal Latin square of order n.

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A366332 Minimum number of diagonal transversals in a semicyclic diagonal Latin square of order 2n+1.

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