

A287648


Maximum number of diagonal transversals in a diagonal Latin square of order n.


10




OFFSET

1,4


COMMENTS

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element.
A diagonal transversal is a transversal that includes exactly one element from the main diagonal and exactly one from the antidiagonal. For squares of odd orders, these elements can coincide at the intersection of the diagonals. (End)
a(11) >= 4828, a(12) >= 24901, a(13) >= 131106, a(14) >= 364596, a(15) >= 389318.  Natalia Makarova, Tomáš Brada, Harry White, Oct 04 2020
a(14) >= 380718, a(20) >= 90010806304, a(21) >= 51162162017, a(22) >= 3227747329246. The number of Dtransversals for orders 20  22 was calculated by a volunteer.  Natalia Makarova, Tomáš Brada, Harry White, Mar 17 2021
a(10) >= 890, a(12) >= 30192, a(14) >= 488792, a(15) >= 4620434, a(17) >= 204995269, a(18) >= 281593874, a(19) >= 11254190082.  Eduard I. Vatutin, Jul 22 2020, updated Mar 09 2022
For most orders n, at least one diagonal Latin square with the maximal number of diagonal transversals has an orthogonal mate and a(n) = A360220(n). Known exceptions: n=6 and n=10.  Eduard I. Vatutin, Feb 17 2023


REFERENCES

J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, and W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics. 1992. Vol. 139. pp. 4349.


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 BOINCbased High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 614. urn:nbn:de:007419730.


EXAMPLE

For example, the diagonal Latin square
0 1 2 3
3 2 1 0
1 0 3 2
2 3 0 1
has 4 diagonal transversals:
0 . . . . 1 . . . . 2 . . . . 3
. . 1 . . . . 0 3 . . . . 2 . .
. . . 2 . . 3 . . 0 . . 1 . . .
. 3 . . 2 . . . . . . 1 . . 0 .
In addition there are 4 other transversals that are not diagonal transversals and are therefore not included here.
The following DLS of order 14 has 364596 diagonal transversals:
0 7 6 11 9 3 4 5 2 12 13 8 10 1
6 1 11 5 10 12 2 3 9 7 4 13 0 8
5 11 2 12 8 1 7 10 0 6 9 3 13 4
13 6 5 3 1 10 9 12 7 0 2 4 8 11
12 3 10 1 4 13 8 6 11 5 0 7 2 9
10 12 1 8 2 5 11 13 4 3 6 0 9 7
9 2 7 0 5 11 6 8 13 4 1 10 3 12
4 13 3 9 6 0 10 7 1 8 12 2 11 5
2 4 9 10 11 6 1 0 8 13 7 12 5 3
1 10 8 13 12 2 5 4 3 9 11 6 7 0
3 5 12 7 13 8 0 1 6 11 10 9 4 2
8 0 13 4 7 9 3 2 12 10 5 11 1 6
7 9 0 6 3 4 13 11 5 2 8 1 12 10
11 8 4 2 0 7 12 9 10 1 3 5 6 13
(End)


CROSSREFS



KEYWORD

nonn,more,hard


AUTHOR



EXTENSIONS



STATUS

approved



