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A308437 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.

%I A308437 #31 Jan 06 2023 15:23:08
%S A308437 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,
%T A308437 1,369,1438,739,436,141,36,6,1,1285,6818,3008,1853,687,193,44,6,1,
%U A308437 4655,27713,12823,7668,3233,1039,268,54,7,1,17073,125830,51619,30902,14731,5164,1518,351,64,7,1
%N A308437 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.
%C A308437 Null tilings (no k X 1 tiles at all) are not counted. _Peter Munn_, May 30 2019
%C A308437 There are A000105(n) free n-ominoes. In a loop over all of them, first consider one fixed representative.
%C A308437 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.
%C A308437 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.
%C A308437 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.
%C A308437 T(n,k) is the number of (partial) tilings of the free n-ominoes with c(i,n,k) straight k-ominoes.
%H A308437 R. J. Mathar, <a href="/A308437/a308437.pdf">Illustrations of n-omino "best" covers with straight k-ominoes.</a>
%F A308437 T(n,1) = A000105(n).
%F A308437 T(n,n) = 1.
%e A308437 The triangle starts with n >= 1, 1 <= k <= n as follows:
%e A308437   1;
%e A308437   1, 1;
%e A308437   2, 4, 1;
%e A308437   5, 8, 4, 1;
%e A308437   12, 35, 18, 4, 1;
%e A308437   35, 89, 61, 22, 5, 1;
%e A308437   108, 425, 206, 97, 28, 5, 1;
%e A308437   369, 1438, 739, 436, 141, 36, 6, 1;
%e A308437   1285, 6818, 3008, 1853, 687, 193, 44, 6, 1;
%e A308437   (...)
%e A308437 From _M. F. Hasler_ and _R. J. Mathar_, May 27 2019: (Start)
%e A308437 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.
%e A308437 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.
%e A308437 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)
%Y A308437 Cf. A000105, A056786, A059573.
%K A308437 nonn,tabl
%O A308437 1,4
%A A308437 _R. J. Mathar_, May 27 2019
%E A308437 NAME improved, _Peter Munn_, May 30 2019