405 research outputs found
The number of directed k-convex polyominoes
We present a new method to obtain the generating functions for directed
convex polyominoes according to several different statistics including: width,
height, size of last column/row and number of corners. This method can be used
to study different families of directed convex polyominoes: symmetric
polyominoes, parallelogram polyominoes. In this paper, we apply our method to
determine the generating function for directed k-convex polyominoes. We show it
is a rational function and we study its asymptotic behavior
Gr\"obner bases of balanced polyominoes
We introduce balanced polyominoes and show that their ideal of inner minors
is a prime ideal and has a quadratic Gr\"obner basis with respect to any
monomial order, and we show that any row or column convex and any tree-like
polyomino is simple and balanced
Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings
This paper provides a bridge between the classical tiling theory and the
complex neighborhood self-assembling situations that exist in practice. The
neighborhood of a position in the plane is the set of coordinates which are
considered adjacent to it. This includes classical neighborhoods of size four,
as well as arbitrarily complex neighborhoods. A generalized tile system
consists of a set of tiles, a neighborhood, and a relation which dictates which
are the "admissible" neighboring tiles of a given tile. Thus, in correctly
formed assemblies, tiles are assigned positions of the plane in accordance to
this relation. We prove that any validly tiled path defined in a given but
arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of
microtiles. A ribbon is a special kind of polyomino, consisting of a
non-self-crossing sequence of tiles on the plane, in which successive tiles
stick along their adjacent edge. Finally, we extend this construction to the
case of traditional tilings, proving that we can simulate
arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving
some of their essential properties.Comment: Submitted to Theoretical Computer Scienc
A closed formula for the number of convex permutominoes
In this paper we determine a closed formula for the number of convex
permutominoes of size n. We reach this goal by providing a recursive generation
of all convex permutominoes of size n+1 from the objects of size n, according
to the ECO method, and then translating this construction into a system of
functional equations satisfied by the generating function of convex
permutominoes. As a consequence we easily obtain also the enumeration of some
classes of convex polyominoes, including stack and directed convex
permutominoes
Covariogram of non-convex sets
The covariogram of a compact set A contained in R^n is the function that to
each x in R^n associates the volume of A intersected with (A+x). Recently it
has been proved that the covariogram determines any planar convex body, in the
class of all convex bodies. We extend the class of sets in which a planar
convex body is determined by its covariogram. Moreover, we prove that there is
no pair of non-congruent planar polyominoes consisting of less than 9 points
that have equal discrete covariogram.Comment: 15 pages, 7 figures, accepted for publication on Mathematik
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