A123234 Number of n X n Latin squares up to row and column permutation (or "RC-equivalence").

1, 1, 1, 4, 16, 1868, 2420400, 66915816462

Offset: 1

Author

Dan Eilers, Oct 06 2006

Keywords

• more
• nice
• nonn

Comments

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 [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.

"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)

Examples

			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.
		

References

• 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)

Links

• B. D. McKay, A. Meynert, W. Myrvold, (2007), Small latin squares, quasigroups, and loops, J. Combin. Designs, 15 (2007), 98-119. doi:10.1002/jcd.20105
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

Cf. A000315, A002724, A058163.