A181502 Triangle read by rows: number of solutions of n queens problem for given n and given maximal size of a connection component in the conflict constellation.

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 8, 4, 0, 0, 0, 0, 0, 0, 64, 24, 4, 0, 0, 0, 0, 0, 0, 248, 80, 16, 8, 0, 0, 0, 0, 0, 0, 172, 484, 36, 32, 0, 0, 0

Offset: 0

Author

Matthias Engelhardt, Oct 30 2010

Keywords

• nonn
• tabl

Comments

Torus solutions, i.e. solutions having an empty conflict constellation, are counted in column 1; this is caused by an interpretation of a queen not engaged in any conflict as an island in the conflict graph. Using the definition strictly, these queens should be removed from the graph and the numbers should appear in column 0, not column 1.

Examples

			Triangle begins:
  0;
  0,  1;
  0,  0, 0;
  0,  0, 0, 0;
  0,  0, 2, 0, 0;
  0, 10, 0, 0, 0, 0;
  0,  0, 0, 0, 4, 0, 0;
  0, 28, 8, 4, 0, 0, 0, 0;
... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two conflicts So the terms for n=4 are 0 (0 solutions for n=4 having 0 conflicts), 0, 2 (the two cited above), 0 and 0. These are members 10 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.

Crossrefs

Cf. A000170, A007705, A181499, A181500, A181501.

Formula

Row sum =A000170 (number of n queens placements)

Column 1 has same values as A007705 (torus n queens solutions)

Column 0 is always zero.

Extensions

Offset corrected by Andrew Howroyd, Dec 31 2017