A181502 -id:A181502

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A181499

Triangle read by rows: number of solutions of n queens problem for given n and given number of conflicts

+10
4

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 0, 0, 8, 4, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 0, 96, 24, 0, 0, 0, 0, 0, 0, 240, 0, 372, 112, 0, 0, 0, 0, 0, 88, 0, 0, 328, 1252, 872, 140, 0, 0, 0, 0, 0, 0, 0, 0, 3016, 5140, 4696, 1316, 32, 0, 0, 0, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Matthias Engelhardt, Table of n, a(n) for n = 0..152 (corrected by Michel Marcus, Jan 19 2019)

M. Engelhardt, Conflicts in the n-queens problem

M. 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)

Column 1 is always zero.

EXAMPLE

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.

CROSSREFS

Cf. A000170, A007705, A181500, A181501, A181502.

KEYWORD

nonn,tabl

AUTHOR

Matthias Engelhardt, Oct 25 2010

STATUS

approved

A181500

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,15

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.

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.

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.

EXAMPLE

Triangle begins:

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.

CROSSREFS

Cf. A181499, A181501, A181502.

KEYWORD

nonn,tabl

AUTHOR

Matthias Engelhardt, Oct 30 2010

EXTENSIONS

Offset corrected by Andrew Howroyd, Dec 31 2017

STATUS

approved

A181501

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

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:

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.

KEYWORD

nonn,tabl

AUTHOR

Matthias Engelhardt, Oct 30 2010

EXTENSIONS

Offset corrected by Andrew Howroyd, Dec 31 2017

STATUS

approved

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