A309088 a(n) is the number of isotopy classes of order n Latin squares that produce a unique determinant.

1, 1, 1, 1, 2, 8, 25

Offset: 1

Author

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 11 2019

Keywords

• nonn
• hard
• more

Comments

We apply every symbol permutation on the representatives of isotopic classes to generate Latin squares of order n and calculate the determinants.

These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.

Examples

			For n=5, the only isotopic class that produces determinants 825, 1875, and 2325 is the one with [[1, 2, 3, 4, 5] [2, 3, 5, 1, 4], [3, 5, 4, 2, 1], [4, 1, 2, 5, 3], [5, 4, 1, 3, 2]] as a representative, and the only isotopic class that produces determinants 1200 and 1575 is the one with [[1, 2, 3, 4, 5], [2, 4, 1, 5, 3], [3, 5, 4, 2, 1], [4, 1, 5, 3, 2], [5, 3, 2, 1, 4]] as a representative.
Therefore, a(5)=2 since there are two isotopic classes that produce determinants that are unique to that isotopic class.
		

Links

• Froylan Maldonado, Sage code
• Brendan McKay, Latin squares
• Brendan McKay, Order 4 isotopic classes
• Brendan McKay, Order 5 isotopic classes
• Brendan McKay, Order 6 isotopic classes
• Brendan McKay, Order 7 isotopic classes
• Index entries for sequences related to determinants

Crossrefs

Cf. A301371, A308853.

Programs

Sage

See Maldonado link.