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

Author

Eduard I. Vatutin, Jun 16 2021

Keywords

• nonn
• more
• hard

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A287647, A287648, A309344, A344105, A345760, A345761, A349199.

Extensions

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

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