187 research outputs found
Hard and Easy Instances of L-Tromino Tilings
We study tilings of regions in the square lattice with L-shaped trominoes.
Deciding the existence of a tiling with L-trominoes for an arbitrary region in
general is NP-complete, nonetheless, we identify restrictions to the problem
where it either remains NP-complete or has a polynomial time algorithm. First,
we characterize the possibility of when an Aztec rectangle and an Aztec diamond
has an L-tromino tiling. Then, we study tilings of arbitrary regions where only
rotations of L-trominoes are available. For this particular case we
show that deciding the existence of a tiling remains NP-complete; yet, if a
region does not contains certain so-called "forbidden polyominoes" as
sub-regions, then there exists a polynomial time algorithm for deciding a
tiling.Comment: Full extended version of LNCS 11355:82-95 (WALCOM 2019
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Polyomino convolutions and tiling problems
We define a convolution operation on the set of polyominoes and use it to
obtain a criterion for a given polyomino not to tile the plane (rotations and
translations allowed). We apply the criterion to several families of
polyominoes, and show that the criterion detects some cases that are not
detectable by generalized coloring arguments.Comment: 8 pages, 8 figures. To appear in \emph{J. of Combin. Theory Ser. A
- β¦