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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.
5
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, 0, 0, 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, 1, 6, 84, 681, 4015, 16149, 46472, 95838, 143657
OFFSET
1,9
COMMENTS
Also number of non-equivalent ways to place k non-attacking wazirs on an n X n board.
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).
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).
Row index starts from n = 1, column index k ranges from 0 to n^2.
T(n, 1) = A008805(n-1); T(n, 2) = A232567(n) for n >= 2; T(n, 3) = A232568(n) for n >= 2;
Into an n X n binary matrix there can be placed maximally A000982(n) = ceiling(n^2/2) pairwise not adjacent 1's.
EXAMPLE
Triangle begins:
1,1;
1,1,1,0,0;
1,3,6,6,3,1,0,0,0,0;
1,3,17,40,62,45,20,4,1,0,0,0,0,0,0,0,0;
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;
...
There are T(3, 2) = 6 non-equivalent binary 3 X 3 matrices with 2 not adjacent 1's (and no other 1's):
[1 0 0] [0 1 0] [1 0 0] [0 1 0] [1 0 1] [1 0 0]
|0 0 0| |0 0 0| |0 1 0| |1 0 0| |0 0 0| |0 0 1|
[0 0 1] [0 1 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0]
KEYWORD
nonn,tabf
AUTHOR
Heinrich Ludwig, Nov 29 2013
STATUS
approved