130 research outputs found
Polyominoes with nearly convex columns: An undirected model
Column-convex polyominoes were introduced in 1950's by Temperley, a
mathematical physicist working on "lattice gases". By now, column-convex
polyominoes are a popular and well-understood model. There exist several
generalizations of column-convex polyominoes; an example is a model called
multi-directed animals. In this paper, we introduce a new sequence of supersets
of column-convex polyominoes. Our model (we call it level m column-subconvex
polyominoes) is defined in a simple way. We focus on the case when cells are
hexagons and we compute the area generating functions for the levels one and
two. Both of those generating functions are complicated q-series, whereas the
area generating function of column-convex polyominoes is a rational function.
The growth constants of level one and level two column-subconvex polyominoes
are 4.319139 and 4.509480, respectively. For comparison, the growth constants
of column-convex polyominoes, multi-directed animals and all polyominoes are
3.863131, 4.587894 and 5.183148, respectively.Comment: 26 pages, 14 figure
Enumeration of generalized polyominoes
As a generalization of polyominoes we consider edge-to-edge connected
nonoverlapping unions of regular -gons. For we determine formulas
for the number of generalized polyominoes consisting of regular
-gons. Additionally we give a table of the numbers for small
and obtained by computer enumeration. We finish with some open problems for
-polyominoes.Comment: 10 pages, 6 figures, 3 table
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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