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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.

%I A181502 #16 Jul 22 2025 08:41:54
%S A181502 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,
%T A181502 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,
%U A181502 0,0,0
%N 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.
%C A181502 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.
%H A181502 M. Engelhardt, <a href="/A181502/b181502.txt">Rows n=0..16 of triangle, flattened</a>
%H A181502 Matthias Engelhardt, <a href="http://nqueens.de/sub/Conflicts.en.html">Conflicts in the n-queens problem</a>
%H A181502 Matthias Engelhardt, <a href="http://nqueens.de/sub/ConflictTables.en.html">Conflict tables for the n-queens problem</a>
%H A181502 M. R. Engelhardt, <a href="http://dx.doi.org/10.1016/j.disc.2007.01.007">A group-based search for solutions of the n-queens problem</a>, Discr. Math., 307 (2007), 2535-2551.
%F A181502 Row sum =A000170 (number of n queens placements)
%F A181502 Column 1 has same values as A007705 (torus n queens solutions)
%F A181502 Column 0 is always zero.
%e A181502 Triangle begins:
%e A181502   0;
%e A181502   0,  1;
%e A181502   0,  0, 0;
%e A181502   0,  0, 0, 0;
%e A181502   0,  0, 2, 0, 0;
%e A181502   0, 10, 0, 0, 0, 0;
%e A181502   0,  0, 0, 0, 4, 0, 0;
%e A181502   0, 28, 8, 4, 0, 0, 0, 0;
%e A181502 ... - _Andrew Howroyd_, Dec 31 2017
%e A181502 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.
%Y A181502 Cf. A000170, A007705, A181499, A181500, A181501.
%K A181502 nonn,tabl
%O A181502 0,13
%A A181502 _Matthias Engelhardt_, Oct 30 2010
%E A181502 Offset corrected by _Andrew Howroyd_, Dec 31 2017