n queen topics

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

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Prepared by Matthias Engelhardt

Mail an Matthias Engelhardt

last change: 2013-04-14

Address of page: http://nqueens.de/sub/CountingResults.en.html