A350585 -id:A350585 - 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

A357514 Minimum number of transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 0, 23, 16, 132

Offset: 1

Eduard I. Vatutin, Oct 01 2022

Orthogonal diagonal Latin squares is a diagonal Latin squares that have at least one orthogonal diagonal mate.
a(10) <= 652, a(11) <= 2091, a(12) <= 6240. - Eduard I. Vatutin, Oct 01 2022, updated Oct 21 2024
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= a(n) <= A287644(n) <= A090741(n). - Eduard I. Vatutin, Feb 17 2023

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, 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. A287644, A287645, A344105, A350585.

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.

A387187 a(n) is the number of distinct numbers of transversals an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 5, 244, 62

Offset: 1

Eduard I. Vatutin, Aug 21 2025

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

			For n=8 the number of transversals that an extended self-orthogonal diagonal Latin square of order 7 may have is 128, 192, 224, 256, or 384. Since there are 3 distinct values, a(8)=5.

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

Cf. A309210, A309598, A309599, A344105, A350585, A383684, A387124.

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

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A357514 Minimum number of 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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A387187 a(n) is the number of distinct numbers of transversals an extended self-orthogonal diagonal Latin square of order n.

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