A181499 -id:A181499 - cp's OEIS frontend

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

Matthias Engelhardt, Oct 30 2010

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.

			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.

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.

Cf. A181499, A181501, A181502.

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.

Offset corrected by Andrew Howroyd, Dec 31 2017

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

Original entry on oeis.org

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

Matthias Engelhardt, Oct 30 2010

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.

			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.

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.

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

Row sum =A000170 (number of n queens placements)

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

Offset corrected by Andrew Howroyd, Dec 31 2017

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.

Original entry on oeis.org

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

Matthias Engelhardt, Oct 30 2010

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.

			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.

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.

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

Row sum =A000170 (number of n queens placements)
Column 1 has same values as A007705 (torus n queens solutions)
Column 0 is always zero.

Offset corrected by Andrew Howroyd, Dec 31 2017

cp's OEIS Frontend

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

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

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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.

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