A181501 Triangle read by rows: number of solutions of n queens problem for given n and given number of connection components of conflict constellation

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 28, 0, 4, 8, 0, 0, 0, 0, 0, 0, 92, 0, 0, 0, 0, 0, 0, 0, 8, 272, 56, 16, 0, 0, 0, 0, 0, 0, 96, 344, 240, 44, 0, 0, 0, 0, 0, 0

Offset

0,13

Comments

The rightmost part of the triangle contains only zeros. As any connection component needs at least two queens, the number of connection components of a solution is always less than or equal to n.

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.

Formula

Row sum =A000170 (number of n queens placements)

Column 0 has same values as A007705 (torus n queens solutions)

Example

Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 2, 0, 0;
  10, 0, 0, 0, 0, 0;
   0, 4, 0, 0, 0, 0, 0;
  28, 0, 4, 8, 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 connection components in the conflicts graph. So, the terms for n=4 are 0, 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

Crossrefs

Cf. A000170, A007705, A181499, A181500, A181502.

Sequence in context: A350793 A181499 A010893 * A213704 A278099 A000171

Adjacent sequences: A181498 A181499 A181500 * A181502 A181503 A181504

Keyword

nonn,tabl

Author

Matthias Engelhardt, Oct 30 2010

Extensions

Offset corrected by Andrew Howroyd, Dec 31 2017

Status

approved