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A232569 Triangle T(n, k) = number of non-equivalent (mod D_4) binary n X n matrices with k pairwise not adjacent 1's; k=0,...,n^2.

%I A232569 #28 Feb 22 2018 14:53:45
%S A232569 1,1,1,1,1,0,0,1,3,6,6,3,1,0,0,0,0,1,3,17,40,62,45,20,4,1,0,0,0,0,0,0,
%T A232569 0,0,1,6,43,210,683,1425,1936,1696,977,366,101,21,5,1,0,0,0,0,0,0,0,0,
%U A232569 0,0,0,0,1,6,84,681,4015,16149,46472,95838,143657
%N A232569 Triangle T(n, k) = number of non-equivalent (mod D_4) binary n X n matrices with k pairwise not adjacent 1's; k=0,...,n^2.
%C A232569 Also number of non-equivalent ways to place k non-attacking wazirs on an n X n board.
%C A232569 Two matrix elements are considered adjacent, if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
%C A232569 Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the corresponding numbers are A232833(n).
%C A232569 Row index starts from n = 1, column index k ranges from 0 to n^2.
%C A232569 T(n, 1) = A008805(n-1); T(n, 2) = A232567(n) for n >= 2; T(n, 3) = A232568(n) for n >= 2;
%C A232569 Into an n X n binary matrix there can be placed maximally A000982(n) = ceiling(n^2/2) pairwise not adjacent 1's.
%H A232569 Heinrich Ludwig, <a href="/A232569/b232569.txt">Rows n = 1..8 of irregular triangle, flattened</a>
%e A232569 Triangle begins:
%e A232569 1,1;
%e A232569 1,1,1,0,0;
%e A232569 1,3,6,6,3,1,0,0,0,0;
%e A232569 1,3,17,40,62,45,20,4,1,0,0,0,0,0,0,0,0;
%e A232569 1,6,43,210,683,1425,1936,1696,977,366,101,21,5,1,0,0,0,0,0,0,0,0,0,0,0,0;
%e A232569 ...
%e A232569 There are T(3, 2) = 6 non-equivalent binary 3 X 3 matrices with 2 not adjacent 1's (and no other 1's):
%e A232569   [1 0 0]   [0 1 0]   [1 0 0]   [0 1 0]   [1 0 1]   [1 0 0]
%e A232569   |0 0 0|   |0 0 0|   |0 1 0|   |1 0 0|   |0 0 0|   |0 0 1|
%e A232569   [0 0 1]   [0 1 0]   [0 0 0]   [0 0 0]   [0 0 0]   [0 0 0]
%Y A232569 Cf. A232567, A232568, A239576, A008805, A000982, A201511, A232833.
%K A232569 nonn,tabf
%O A232569 1,9
%A A232569 _Heinrich Ludwig_, Nov 29 2013