A123234 - OEIS

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

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OFFSET

1,4

COMMENTS

"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)

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

Table of n, a(n) for n=1..8.

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

EXAMPLE

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.