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%I #15 Jul 21 2020 02:48:22
%S 1,1,1,4,16,1868,2420400,66915816462
%N Number of n X n Latin squares up to row and column permutation (or "RC-equivalence").
%C _Brendan McKay_ writes: (Start)
%C "It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD [see link below]. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.
%C "Is(L) is the autotopism group. Also define the group RC(L) of all autotopisms for which the symbols component is the identity. For any Latin square L we have:
%C "The isotopy class containing L contains (n!)^3/|Is(L)| squares.
%C "The RC-equivalence class containing L contains (n!)^2/|RC(L)| squares.
%C "If L and L' are isotopic then |RC(L)| = |RC(L')|. Therefore the number of RC-equivalence classes in the isotopy class of L is n!*|RC(L)|/|Is(L)|. I modified an existing program slightly to find |RC(L)|/|Is(L)|. and applied it to one square from each isotopy class. The sum of n!*|RC(L)|/|Is(L)| is the total number of RC-equivalence classes. " (End)
%D Dan R. Eilers, Phil A. Sallee, The number of Latin squares up to row and column permutation, Poster Session, Harvey Mudd College Mathematics Conference on Enumerative Combinatorics (2006) (for terms 1 to 7)
%D Brendan D. McKay, private communication (2006) (for term 8)
%H B. D. McKay, A. Meynert, W. Myrvold, (2007), <a href="http://users.cecs.anu.edu.au/~bdm/papers/ls_final.pdf">Small latin squares, quasigroups, and loops</a>, J. Combin. Designs, 15 (2007), 98-119. <a href="https://doi.org/10.1002/jcd.20105">doi:10.1002/jcd.20105</a>
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%e 01234 => 20413 => 01234
%e 13042 => 01234 => 14320
%e 24310 => 32041 => 20413
%e 30421 => 43102 => 32041
%e 42103 => 14320 => 43102
%e The first square is transformed by permuting columns; the 2nd square is transformed by permuting rows.
%e Both the first and 3rd square are in reduced form, so are considered equivalent by row/col permutation.
%Y Cf. A000315, A002724, A058163.
%K more,nice,nonn
%O 1,4
%A _Dan Eilers_, Oct 06 2006
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