A344105 a(n) is the number of distinct numbers of transversals of order n diagonal Latin squares.

1, 0, 0, 1, 2, 1, 32, 73, 406

Offset: 1

Author

Eduard I. Vatutin, Jun 22 2021

Keywords

• nonn
• more
• hard

Comments

a(n) <= A287644(n) - A287645(n) + 1.

a(n) <= A287764(n).

Diagonal Latin squares are a special case of Latin squares, so a(n) <= A309344(n).

a(10) >= 459, a(11) >= 6437, a(12) >= 23707, a(13) >= 75891, a(14) >= 290681. - Eduard I. Vatutin, Oct 29 2021, updated Mar 01 2025

For all spectra of even orders all known values included in them are divisible by 2. For all spectra of orders n=6, n=10 and n=14 (and probably for all n=4k+2) all known values included in the corresponding spectra are divisible by 4. This leads to the following hypothesis: a(2k) <= (A287644(2k) - A287645(2k) + 2)/2 and a(4k+2) <= (A287644(4k+2) - A287645(4k+2) + 4)/4, where w(n) = A287644(n) - A287645(n) + 1 is a width of corresponding spectra and (w(n)+1)/2 is done to round the result of the division up. - Eduard I. Vatutin, Mar 21 2022

Examples

			For n=7 the number of transversals that a diagonal Latin square of order 7 may have is 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 41, 43, 45, 47, 55, or 133. Since there are 32 distinct values, a(7)=32.
		

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, About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9 (in Russian).
• 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, Graphical representation of the spectra.
• Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
• 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.
• 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, 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)
• E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, and 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.
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A287644, A287645, A287764, A309344, A345370, A345760, A345761.

Extensions

a(8) added by Eduard I. Vatutin, Jul 14 2021

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