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Counting results for the n Queens Problem
For counting the solutions, much information on the problem is contained
in OnLine Encyclopedia of Integer Sequences. I think it is not meaningful to repeat these sequences here.
Instead, you find links to special sequences by the following tables.
Numbers of the sequences will be given here if it is in a table for direct comparision,
if they are too special for the OEIS, or if the are not yet validated enough.
The following tables are organized by the different base sets and by the symmetries. Base sets are:
  all permutations
  solutions to the normal n queens problem
  solutions to the torus n queens problem
  permutations having at most two queens on a (anti)diagonal
Permutations as base sets (i.e. rooks instead of queens)
Normal queens
 
Permutations 
Dihedral group, i.e. reflect and rotate 
Congruencies on the torus 
Similarity 
Regular affine mappings 
All affine mappings 
all 
 classic case 
A000170 
A002562 
A062164 
A062165 
 
 
central symmetry (i.e. for rotation of 180°) 
A032522 
 
 
 
 
 
rotational symmetry (90°) 
A033148 
 
 
 
 
 
shift symmetries 
 
 
 
 
 
 
other symmetries 
 
 
 
 
 
 
Torus queens
 
Permutations 
Dihedral group, i.e. reflect and rotate 
Congruencies on the torus 
Similarity 
Regular affine mappings 
All affine mappings 
all 
A007705 
 
A053994 
A062166 
 
 
central symmetry (i.e. for rotation of 180°) 
 
 
 
 
 
 
rotational symmetry (90°) 
 
 
 
 
 
 
shift symmetries 
 
 
 
 
 
 
other symmetries 
 
 
A054500
A054501
A054502

 
 
 
Permutations having at most two queens on a diagonal
 
Permutations 
Dihedral group, i.e. reflect and rotate 
Congruencies on the torus 
Similarity 
Regular affine mappings 
All affine mappings 
all 
 
 
A062167 
A062168 
 
 
central symmetry (i.e. for rotation of 180°) 
 
 
 
 
 
 
rotational symmetry (90°) 
 
 
 
 
 
 
shift symmetries 
 
 
 
 
 
 
other symmetries 
 
 
 
 
 
 
As you see, there are still many unknown sequences which
are also of some interest.
Preliminary results
Number of orbits of permutations, under action of similarity (n=1 .. 11): 1,1,1,2,4,10,12,80,232,2616,8513
Number of affected orbits of permutations, under action of affinity (n=1 .. 11): 1,1,1,2,2,10,7,42,92,1294,1825
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