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# Counting results for the n Queens Problem

For counting the solutions, much information on the problem is contained in On-Line 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)

 - Permutations Dihedral group, i.e. reflect and rotate Congruencies on the torus Similarity Regular affine mappings All affine mappings all A000142 A000903 A006841 Permutations under similarity - Permutations under affinity central symmetry (i.e. for rotation of 180°) A037223 - - - - - rotational symmetry (90°) A037224 - - - - - shift symmetries - - - - - - other symmetries Fixed by composition of rotation and reflection  A000085 - - - - -

## 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

Prepared by Matthias Engelhardt

last change: 2013-04-14