|
|
A287645
|
|
Minimum number of transversals in a diagonal Latin square of order n.
|
|
7
|
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A287644(n) <= A090741(n).
A lower bound for odd n is A091323((n-1)/2) <= a(n). (End)
By definition, the main diagonal and antidiagonal of a diagonal Latin square are transversals, so a(n)>=2 for all n>=4 (the two diagonals are the same in the order 1 square and there are no diagonal Latin squares of orders 2 or 3). - Eduard I. Vatutin, Jun 13 2021
All cyclic diagonal Latin squares are diagonal Latin squares, so a(n) <= A348212((n-1)/2) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021
a(10) <= 144, a(11) <= 1721, a(12) <= 448, a(13) <= 43093, a(14) <= 65432, a(15) <= 215721, a(16) <= 7465984. - Eduard I. Vatutin, Mar 11 2021, updated Jul 10 2023
|
|
LINKS
|
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. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.
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)
|
|
EXAMPLE
|
For example, diagonal Latin square
0 1 2 3
3 2 1 0
1 0 3 2
2 3 0 1
has 4 diagonal transversals (see A287648)
0 . . . . 1 . . . . 2 . . . . 3
. . 1 . . . . 0 3 . . . . 2 . .
. . . 2 . . 3 . . 0 . . 1 . . .
. 3 . . 2 . . . . . . 1 . . 0 .
and 4 not diagonal transversals
0 . . . . 1 . . . . 2 . . . . 3
. 2 . . 3 . . . . . . 0 . . 1 .
. . 3 . . . . 2 1 . . . . 0 . .
. . . 1 . . 0 . . 3 . . 2 . . .
total 8 transversals. (End)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,more,hard
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|