A181500 Triangle read by rows: number of solutions of n queens problem for given n and given number of queens engaged in conflicts.

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 28, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 8, 32, 48, 32

Offset: 0

Author

Matthias Engelhardt, Oct 30 2010

Keywords

• nonn
• tabl

Comments

Schlude and Specker investigate if it is possible to set n-1 non-attacking queens on an n X n toroidal chessboard. That is equivalent to searching for normal (i.e., non-toroidal) solutions of 3 engaged queens. In this case, one of the three queens has conflicts with both other queens. If you remove this queen, you get a setting of n-1 queens without conflicts, i.e., a toroidal solution.

Examples

			Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 0, 0, 2;
  10, 0, 0, 0, 0, 0;
   0, 0, 0, 0, 4, 0,  0;
  28, 0, 0, 0, 0, 0, 12, 0;
... - _Andrew Howroyd_, Dec 31 2017
For n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. For both solutions, all 4 queens are engaged in conflicts. So the terms for n=4 are 0 (0 solutions for n=4 having 0 engaged queens), 0, 0, 0 and 2 (the two cited above). These are members 11 to 15 of the sequence.
		

Links

• M. Engelhardt, Rows n=0..16 of triangle, flattened
• Matthias Engelhardt, Conflicts in the n-queens problem
• Matthias Engelhardt, Conflict tables for the n-queens problem
• M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.
• Konrad Schlude and Ernst Specker, Zum Problem der Damen auf dem Torus, Technical Report 412, Computer Science Department ETH Zurich, 2003.

Crossrefs

Cf. A181499, A181501, A181502.

Formula

Row sum = A000170 (number of n-queen placements).

Column 0 has same values as A007705 (torus n-queen solutions).

Columns 1 and 2 are always zero.

Column 3 counts solutions of the special "Schlude-Specker" situation.

Extensions

Offset corrected by Andrew Howroyd, Dec 31 2017