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

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

Offset

1,4

Comments

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

Links

Table of n, a(n) for n=1..9.

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

Formula

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

Example

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

Crossrefs

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

Sequence in context: A111179 A178173 A058891 * A184945 A058343 A267407

Adjacent sequences: A274168 A274169 A274170 * A274172 A274173 A274174

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Jul 07 2016

Extensions

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

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

Status

approved