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

LINKS

Heinrich Ludwig, Rows n = 1..8 of irregular triangle, flattened

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]

CROSSREFS

Cf. A232567, A232568, A239576, A008805, A000982, A201511, A232833.

Sequence in context: A086727 A021277 A245215 * A228022 A016662 A258271

Adjacent sequences:  A232566 A232567 A232568 * A232570 A232571 A232572

KEYWORD

nonn,tabf

AUTHOR

Heinrich Ludwig, Nov 29 2013

STATUS

approved

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Last modified January 17 14:12 EST 2019. Contains 319225 sequences. (Running on oeis4.)