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a(n) is the number of isotopy classes of order n Latin squares that produce a unique determinant.
5

%I #28 Oct 20 2021 07:51:19

%S 1,1,1,1,2,8,25

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

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

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

%H Froylan Maldonado, <a href="/A309088/a309088.sage.txt">Sage code</a>

%H Brendan McKay, <a href="https://users.cecs.anu.edu.au/~bdm/data/latin.html">Latin squares</a>

%H Brendan McKay, <a href="/A309088/a309088.txt">Order 4 isotopic classes</a>

%H Brendan McKay, <a href="/A309088/a309088_1.txt">Order 5 isotopic classes</a>

%H Brendan McKay, <a href="/A309088/a309088_2.txt">Order 6 isotopic classes</a>

%H Brendan McKay, <a href="/A309088/a309088_3.txt">Order 7 isotopic classes</a>

%H <a href="/index/De#determinants">Index entries for sequences related to determinants</a>

%e 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.

%e Therefore, a(5)=2 since there are two isotopic classes that produce determinants that are unique to that isotopic class.

%o (Sage) See Maldonado link.

%Y Cf. A301371, A308853.

%K nonn,hard,more

%O 1,5

%A _Alvaro R. Belmonte_, _Eugene Fiorini_, _Peterson Lenard_, _Froylan Maldonado_, _Sabrina Traver_, _Wing Hong Tony Wong_, Jul 11 2019