

A287695


Maximum number of diagonal Latin squares with the first row in ascending order that can be orthogonal to a given diagonal Latin square of order n.


5




OFFSET

1,7


COMMENTS

A Latin square is normalized if in the first row elements come in increasing order. Any diagonal Latin square orthogonal to a given one can be normalized by renaming its elements (which does not break diagonality and orthogonality).  Max Alekseyev, Dec 07 2019
For all orders n>3 there are diagonal Latin squares without orthogonal mates (also known as bachelor squares), so the minimum number of diagonal Latin squares that can be orthogonal to the same diagonal Latin square is zero. For order n=1 the single square is orthogonal to itself. For n=2 and n=3 diagonal Latin squares do not exist (see A274171). For n=6 orthogonal diagonal Latin squares do not exist (see A305571), so a(6)=0.  Eduard I. Vatutin, May 03 2021
a(12) >= 3855983322. The result belongs to DLS, which has 30192 diagonal transversals. Calculations performed by a volunteer.  Natalia Makarova, Tomáš Brada, Nov 11 2021
a(16) >= 1658880, a(17) >= 2453352, a(18) >= 96, a(19) >= 1383, a(20) >= 995328, a(21) >= 995328, a(22) >= 432000, a(23) >= 525, a(24) >= 345600, a(25) >= 345600, a(26) >= 48, a(27) >= 345600, a(28) >= 663552, a(29) >= 663552, a(30) >= 40320. For values up to a(100), see the specified link "New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square".  Natalia Makarova, Alex Chernov, Harry White, Dec 06 2021


LINKS



EXAMPLE

One of the best existing diagonal Latin squares of order 7
0 1 2 3 4 5 6
2 3 1 5 6 4 0
5 6 4 0 1 2 3
4 0 6 2 3 1 5
6 2 0 1 5 3 4
1 5 3 4 0 6 2
3 4 5 6 2 0 1
has 3 orthogonal mates
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6
5 6 4 0 1 2 3 3 4 5 6 2 0 1 6 2 0 1 5 3 4
1 5 3 4 0 6 2 4 0 6 2 3 1 5 3 4 5 6 2 0 1
6 2 0 1 5 3 4 2 3 1 5 6 4 0 1 5 3 4 0 6 2
3 4 5 6 2 0 1 5 6 4 0 1 2 3 2 3 1 5 6 4 0
2 3 1 5 6 4 0 6 2 0 1 5 3 4 4 0 6 2 3 1 5
4 0 6 2 3 1 5 1 5 3 4 0 6 2 5 6 4 0 1 2 3
so a(7)=3. (End)


CROSSREFS



KEYWORD

nonn,more,hard


AUTHOR



EXTENSIONS



STATUS

approved



