A309283 -id:A309283 - cp's OEIS frontend

A287764 Number of main classes of diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 972, 4873096, 3292326155394

Offset: 1

Eduard I. Vatutin, May 31 2017

A. D. Belyshev, Discussion about properties of diagonal Latin squares (in Russian)
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)
E. I. Vatutin, A. D. Belyshev, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
Eduard Vatutin, Alexey Belyshev, Stepan Kochemazov, Oleg Zaikin, Natalia Nikitina, Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing, Russian Supercomputing Days (Суперкомпьютерные дни в России), 2018.
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, 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. Third independent confirmation of the previously calculated value a(9) (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)
Index entries for sequences related to Latin squares and rectangles

Cf. A003090, A309283, A274171.

a(9) from Eduard I. Vatutin, Jul 06 2019

A274171 Number of diagonal Latin squares of order n with the first row in order.

Original entry on oeis.org

1, 0, 0, 2, 8, 128, 171200, 7447587840, 5056994653507584

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Sep 29 2020

			The a(4) = 2 diagonal Latin squares are:
   0 1 2 3   0 1 2 3
   2 3 0 1   3 2 1 0
   3 2 1 0   1 0 3 2
   1 0 3 2   2 3 0 1
.
The a(5) = 8 diagonal Latin squares are:
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   1 3 4 2 0   1 4 3 0 2   2 3 4 0 1   2 4 1 0 3
   4 2 1 0 3   3 2 1 4 0   4 0 1 2 3   4 0 3 2 1
   2 0 3 4 1   4 3 0 2 1   1 2 3 4 0   3 2 4 1 0
   3 4 0 1 2   2 0 4 1 3   3 4 0 1 2   1 3 0 4 2
.
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   3 4 0 1 2   3 4 1 2 0   4 2 0 1 3   4 2 3 0 1
   1 2 3 4 0   4 2 3 0 1   1 4 3 2 0   3 4 1 2 0
   4 0 1 2 3   2 0 4 1 3   3 0 1 4 2   1 3 0 4 2
   2 3 4 0 1   1 3 0 4 2   2 3 4 0 1   2 0 4 1 3

S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.
S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
Eduard I. Vatutin, a(9) value fixed after
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S.Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39 (in Russian).
E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157, (in Russian).
Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., and Titov V.S., Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
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)
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000479, A274806, A287764, A309283.

a(n) = A274806(n)/n!.

a(9) added from Vatutin et al. (2016) by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A274806 Number of diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 960, 92160, 862848000, 300286741708800, 1835082219864832081920

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Oct 05 2020

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian).
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian).
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39, (in Russian).
E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157, (in Russian).
Eduard I. Vatutin, a(9) value fixed
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
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)
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000479, A274171, A287764, A309283.

a(n) = A274171(n) * n!.

a(9) from Vatutin et al. (2016) added by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A337302 Number of X-based filling of diagonals in a diagonal Latin square of order n with the main diagonal in ascending order.

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 80, 80, 4752, 4752, 440192, 440192, 59245120, 59245120, 10930514688, 10930514688, 2649865335040, 2649865335040, 817154768973824, 817154768973824, 312426715251262464, 312426715251262464, 145060238642780180480, 145060238642780180480

Offset: 0

Eduard I. Vatutin, Aug 22 2020

nonn

Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.

For all t > 0, a(2*t) = a(2*t+1).

			For n=4 there are 4 different X-based fillings of diagonals with main diagonal fixed to [0 1 2 3]:
   0 . . 1   0 . . 1   0 . . 2   0 . . 2
   . 1 0 .   . 1 3 .   . 1 0 .   . 1 3 .
   . 3 2 .   . 0 2 .   . 3 2 .   . 0 2 .
   2 . . 3   2 . . 3   1 . . 3   1 . . 3

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
E. I. Vatutin, About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15 (in Russian).
E. I. Vatutin, About the a(2*t)=a(2*t+1) equality (in Russian).
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A000316, A309283, A274171, A337303.

a(n) = A337303(n)/n!.

a(n) = A000316(floor(n/2)). - Andrew Howroyd and Eduard I. Vatutin, Oct 08 2020

More terms from Alois P. Heinz, Oct 08 2020

a(0)=1 prepended by Andrew Howroyd, Oct 09 2020

A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 1, 0, 0, 96, 480, 57600, 403200, 191600640, 1724405760, 1597368729600, 17571056025600, 28378507272192000, 368920594538496000, 952903592436341145600, 14293553886545117184000, 55442575636536644075520000, 942523785821122949283840000, 5231730206388249282710863872000

Offset: 0

Eduard I. Vatutin, Aug 22 2020

nonn

Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.

			One of the 96 X-based fillings of diagonals of a diagonal Latin square for order n=4:
1 . . 0
. 0 1 .
. 3 2 .
2 . . 3

Andrew Howroyd, Table of n, a(n) for n = 0..100
S. Kochemazov, O. Zaikin, E. Vatutin E., and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
E. I. Vatutin, About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15 (in Russian).
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A000316, A000459, A309283, A274171, A337302.

\\ here b(n) is A000459.
b(n) = {sum(m=0, n, sum(k=0, n-m, (-1)^k * binomial(n, k) * binomial(n-k, m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); }
a(n) = {2^(n\2) * b(n\2) * n!} \\ Andrew Howroyd, Mar 26 2023

a(n) = A337302(n)*n!.

a(n) = n!*A000316(floor(n/2)). - Andrew Howroyd, Mar 26 2023

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Mar 26 2023

A338084 Number of equivalence classes of X-based filling of diagonals in a diagonal Latin square of order 2n (or 2n+1).

Original entry on oeis.org

1, 0, 2, 3, 20, 67, 596

Offset: 0

Eduard I. Vatutin, Oct 08 2020

Supplemental for A309283.

The number of solutions in an equivalence class with the main diagonal in ascending order is at most 4*2^n*n!. This maximum is only achieved for n >= 5. - Andrew Howroyd, Mar 27 2023

			From _Andrew Howroyd_, Mar 27 2023: (Start)
For n = 5, the following is an example solution in an equivalence class of maximum size. The second square shows the effect of swapping the two diagonals and renumbering so that the main diagonal is still in ascending order.
   0 . . . . . . . . 1    0 . . . . . . . . 1
   . 1 . . . . . . 0 .    . 1 . . . . . . 0 .
   . . 2 . . . . 3 . .    . . 2 . . . . 3 . .
   . . . 3 . . 2 . . .    . . . 3 . . 2 . . .
   . . . . 4 6 . . . .    . . . . 4 9 . . . .
   . . . . 7 5 . . . .    . . . . 6 5 . . . .
   . . . 5 . . 6 . . .    . . . 4 . . 6 . . .
   . . 8 . . . . 7 . .    . . 5 . . . . 7 . .
   . 9 . . . . . . 8 .    . 7 . . . . . . 8 .
   4 . . . . . . . . 9    8 . . . . . . . . 9
(End)

E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).

Cf. A000316, A309283.

a(n) >= A000316(n) / (4*2^n*n!). - Andrew Howroyd, Mar 27 2023

cp's OEIS Frontend

A287764 Number of main classes of diagonal Latin squares of order n.

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A274171 Number of diagonal Latin squares of order n with the first row in order.

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A274806 Number of diagonal Latin squares of order n.

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A337302 Number of X-based filling of diagonals in a diagonal Latin square of order n with the main diagonal in ascending order.

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A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

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A338084 Number of equivalence classes of X-based filling of diagonals in a diagonal Latin square of order 2n (or 2n+1).

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