9 research outputs found
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
A benchmark set for the reconstruction of hv-convex discrete sets
AbstractIn this paper we summarize the most important generation methods developed for the subclasses of hv-convex discrete sets. We also present some new generation techniques to complement the former ones thus making it possible to design a complete benchmark set for testing the performance of reconstruction algorithms on the class of hv-convex discrete sets and its subclasses. By using this benchmark set the paper also collects several statistics on hv-convex discrete sets, which are of great importance in the analysis of algorithms for reconstructing such kinds of discrete sets
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
A family of classes in nested chain abacus and related generating functions
Abacus model has been employed widely to represent partitions for any positive integer. However, no study has been carried out to develop connected beads of abacus in graphical representation for discrete objects. To resolve this connectedness problem this study is oriented in characterising n - connected objects knows as n connected ominoes, which then generate nested chain abacus. Furthermore, the theoretical conceptual properties for the nested chain abacus are being formulated. Along the construction, three different types of transformation are being created that are essential in building a family of classes. To enhance further, based on theses classes, generating functions are also being formulated by employing enumeration of combinatorial objects (ECO). In ECO method, each object is obtained from smaller object by making some local expansions. These local expansions are described in a simple way by a succession rule which can be translated into a function equation for the generating function. In summary, this stud has succeeded in producing novel graphical representation of nested chain abacus, which can be applied in tiling finite grid
A framework for generating some discrete sets with disjoint components by using uniform distributions
AbstractDiscrete tomography deals with the reconstruction of discrete sets from few projections. Assuming that the set to be reconstructed belongs to a certain class of discrete sets with some geometrical properties is a commonly used technique to reduce the number of possibly many different solutions of the same reconstruction problem. The average performance of reconstruction algorithms are often tested on such classes by choosing elements of a given class from uniform random distributions. This paper presents a general framework for generating discrete sets with disjoint connected components using uniform distributions. Especially, the uniform random generation of hv-convex discrete sets and Q-convex discrete sets according to the size of the minimal bounding rectangle are discussed
On directed-convex polyominoes in a rectangle
We provide bijective proofs for the number of directed-convex polyominoes having a fixed number of rows and columns in two ways: by means of the ECO method, and through a correspondence with the set of 2-colored Grand-Motzkin paths