165 research outputs found
Staircase polygons: moments of diagonal lengths and column heights
We consider staircase polygons, counted by perimeter and sums of k-th powers
of their diagonal lengths, k being a positive integer. We derive limit
distributions for these parameters in the limit of large perimeter and compare
the results to Monte-Carlo simulations of self-avoiding polygons. We also
analyse staircase polygons, counted by width and sums of powers of their column
heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity:
An International Workshop On Statistical Mechanics And Combinatorics, 10-15
July 2005, Queensland, Australi
Perimeter Generating Functions For The Mean-Squared Radius Of Gyration Of Convex Polygons
We have derived long series expansions for the perimeter generating functions
of the radius of gyration of various polygons with a convexity constraint.
Using the series we numerically find simple (algebraic) exact solutions for the
generating functions. In all cases the size exponent .Comment: 8 pages, 1 figur
Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints
In this thesis, we consider the problem of characterizing and enumerating
sets of polyominoes described in terms of some constraints, defined either by
convexity or by pattern containment. We are interested in a well known subclass
of convex polyominoes, the k-convex polyominoes for which the enumeration
according to the semi-perimeter is known only for k=1,2. We obtain, from a
recursive decomposition, the generating function of the class of k-convex
parallelogram polyominoes, which turns out to be rational. Noting that this
generating function can be expressed in terms of the Fibonacci polynomials, we
describe a bijection between the class of k-parallelogram polyominoes and the
class of planted planar trees having height less than k+3. In the second part
of the thesis we examine the notion of pattern avoidance, which has been
extensively studied for permutations. We introduce the concept of pattern
avoidance in the context of matrices, more precisely permutation matrices and
polyomino matrices. We present definitions analogous to those given for
permutations and in particular we define polyomino classes, i.e. sets downward
closed with respect to the containment relation. So, the study of the old and
new properties of the redefined sets of objects has not only become
interesting, but it has also suggested the study of the associated poset. In
both approaches our results can be used to treat open problems related to
polyominoes as well as other combinatorial objects.Comment: PhD thesi
On the enumeration of permutominoes
Although the exact counting and enumeration of polyominoes remain challenging open problems, several positive results were achieved for special classes of polyominoes. We give an algorithm for direct enumeration of permutominoes by size, or, equivalently, for the enumeration of grid orthogonal polygons. We show how the construction technique allows us to derive a simple characterization of the class of convex permutominoes, which has been extensively investigated. The approach extends to other classes, such as the row convex and the directed convex permutominoes.Fondo Europeo de Desarrollo RegionalFundação para a Ciência e a Tecnologi
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