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.

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

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..66.

R. J. Mathar, Illustrations of n-omino "best" covers with straight k-ominoes.

Formula

T(n,1) = A000105(n).

T(n,n) = 1.

Example

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;
  (...)
From M. F. Hasler and R. J. Mathar, May 27 2019: (Start)
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)

Crossrefs

Cf. A000105, A056786, A059573.

Sequence in context: A177985 A198512 A059573 * A201283 A283739 A080427

Adjacent sequences: A308434 A308435 A308436 * A308438 A308439 A308440

Keyword

nonn,tabl

Author

R. J. Mathar, May 27 2019

Extensions

NAME improved, Peter Munn, May 30 2019

Status

approved