1,4

Brendan McKay writes: (Start)

"It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD: http://cs.anu.edu.au/~bdm/papers/ls_final.pdf. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.

"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:

"The isotopy class containing L contains (n!)^3/|Is(L)| squares.

"The RC-equivalence class containing L contains (n!)^2/|RC(L)| squares.

"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)

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)

Brendan D. McKay, private communication (2006) (for term 8)

01234 => 20413 => 01234

13042 => 01234 => 14320

24310 => 32041 => 20413

30421 => 43102 => 32041

42103 => 14320 => 43102

The first square is transformed by permuting columns; the 2nd square is transformed by permuting rows.

Both the first and 3rd square are in reduced form, so are considered equivalent by row/col permutation.

Cf. A000315, A002724, A058163.

Adjacent sequences: A123231 A123232 A123233 this_sequence A123235 A123236 A123237

Sequence in context: A013006 A013001 A013179 this_sequence A116573 A063625 A057596

more,nice,nonn

Dan Eilers (dan(AT)irvine.com), Oct 06 2006