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A308437
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Triangle read by rows: T(n,k) = number of ways, summed over the free n-ominoes, that an n-omino with an assigned orientation can be maximally (partially) covered by k X 1 tiles.
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2
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1, 1, 1, 2, 4, 1, 5, 8, 4, 1, 12, 35, 18, 4, 1, 35, 89, 61, 22, 5, 1, 108, 425, 206, 97, 28, 5, 1, 369, 1438, 739, 436, 141, 36, 6, 1, 1285, 6818, 3008, 1853, 687, 193, 44, 6, 1, 4655, 27713, 12823, 7668, 3233, 1039, 268, 54, 7, 1, 17073, 125830, 51619, 30902, 14731, 5164, 1518, 351, 64, 7, 1
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OFFSET
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1,4
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COMMENTS
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Null tilings (no k X 1 tiles at all) are not counted. Peter Munn, May 30 2019
There are A000105(n) free n-ominoes. In a loop over all of them, first consider one fixed representative.
Consider the straight k-ominoes (in horizontal or vertical alignments commensurate with the grid of the n-omino), and let c(i,n,k) be the maximum number of straight k-ominoes in any mixture of vertical-horizontal alignments that can be placed inside the i-th n-omino such that no k-ominoes overlap and such that all cells of the k-ominoes are cells of the n-omino.
Obviously c(i,n,k) <= floor(n/k): The coverage by a set of fixed k-ominoes is always incomplete if k is not a divisor of n.
Count all configurations with the number of c(i,n,k) k-ominos in the representative. Configurations with distinct multisets of k-ominoes are considered distinct, even if rotations or flips of the (partially) covered n-omino may exist that map these onto others.
T(n,k) is the number of (partial) tilings of the free n-ominoes with c(i,n,k) straight k-ominoes.
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LINKS
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FORMULA
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T(n,n) = 1.
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EXAMPLE
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The triangle starts with n >= 1, 1 <= k <= n as follows:
1;
1, 1;
2, 4, 1;
5, 8, 4, 1;
12, 35, 18, 4, 1;
35, 89, 61, 22, 5, 1;
108, 425, 206, 97, 28, 5, 1;
369, 1438, 739, 436, 141, 36, 6, 1;
1285, 6818, 3008, 1853, 687, 193, 44, 6, 1;
(...)
We have T(n,1) = A000105(n) which is the number of different inequivalent n-ominoes, and each one can be maximally filled in exactly one (trivial) way with 1 X 1 monominoes.
We have T(n,n) = 1 because only the straight n X 1 polyomino can be filled in the required way, namely with only straight n-ominoes.
T(3,2) = 4 counts 2 ways of placing a domino into the straight tromino (the two ends of the tromino considered distinct) and 2 ways of placing a domino into the L-tromino (again the two variants obtained by flipping along the diagonal considered distinct). (End)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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