Results 1  10
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18
New Enumerative Results on TwoDimensional Directed Animals
, 1998
"... We list several open problems concerning the enumeration of directed animals on twodimensional lattices. We show that most of these problems are special cases of two central problems: calculating the position generating function and the perimeter and area generating function for square lattice anim ..."
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Cited by 21 (4 self)
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We list several open problems concerning the enumeration of directed animals on twodimensional lattices. We show that most of these problems are special cases of two central problems: calculating the position generating function and the perimeter and area generating function for square lattice animals. We propose a possible direction for solving these two problems: we extend Dhar's correspondence between hard particle gas models and enumeration of animals according to the area, and show that each of the main two generating functions is, essentially, the density of a onedimensional gas model given by the stationary distribution of a probabilistic transition. We are able to compute the density of certain stationary distributions. We thus obtain new bivariate generating functions for directed animals on the square and triangular lattices. We derive from these results the generating functions for animals on the decorated square and triangular lattices, as well as the average number of lo...
Lattice Animals and Heaps of Dimers
 Discrete Math
, 2003
"... The general quest of this paper is the search for new classes of square lattice animals that are both large and exactly enumerable. The starting point is a bijection between... ..."
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Cited by 20 (5 self)
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The general quest of this paper is the search for new classes of square lattice animals that are both large and exactly enumerable. The starting point is a bijection between...
Probabilistic analysis of columnconvex and directed diagonallyconvex animals II: trajectories and shapes.
 Random Structures & Algorithms
, 2000
"... Using some generating functions and limit theorems, we derive a stochastic description of the trajectories and shapes of columnconvex and directed diagonallyconvex animals for given large area. The limiting processes are characterized by Brownian motions and the animals thickness by some discrete ..."
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Cited by 9 (7 self)
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Using some generating functions and limit theorems, we derive a stochastic description of the trajectories and shapes of columnconvex and directed diagonallyconvex animals for given large area. The limiting processes are characterized by Brownian motions and the animals thickness by some discrete distributions, the maximum of which is also analyzed. A simulated realization is constructed for both kind of animals. 1 Introduction Among combinatorial structures, the animals (also called polyominoes) constitute a quite interesting subeld. An animal is a set of points N N such that every point of the animal can be reached from another point by a sequence of steps in the lattice plane. Usually animals are considered up to a translation. Animals have already been the subject of a large literature: see for instance Viennot [24] for denitions and a nice survey, BousquetMelou [5] for a complete combinatorial approach, Bender [2] for general convex polyominoes analysis and Part I (Loucha...
Critical and multicritical semirandom (1 + d)dimensional lattices and hard objects in d dimensions
 J. Phys. A Math. Gen
, 2002
"... We investigate models of (1+d)D Lorentzian semirandom lattices with one random (spacelike) direction and d regular (timelike) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an ex ..."
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Cited by 7 (2 self)
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We investigate models of (1+d)D Lorentzian semirandom lattices with one random (spacelike) direction and d regular (timelike) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)D Lorentzian surfaces, with fractal dimensions dF = k+1, k = 1, 2, 3,..., as well as a new model of (1+2)D critical tetrahedral complexes, with fractal dimension dF = 12/5. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1+d)D Lorentzian lattices and directedsite lattice animals in (1 + d) dimensions.
Enumeration of Directed Animals on an Infinite Family of Lattices
, 1996
"... We prove algebraic equations satisfied by the area generating function for directed animals on an infinite family of regular, nonplanar, twodimensional graphs. 1 Introduction A directed animal A on an oriented graph having an origin O is a finite set of sites containing O such that each point of ..."
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Cited by 6 (0 self)
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We prove algebraic equations satisfied by the area generating function for directed animals on an infinite family of regular, nonplanar, twodimensional graphs. 1 Introduction A directed animal A on an oriented graph having an origin O is a finite set of sites containing O such that each point of A is connected to O through an oriented path of the graph having all its vertices in A. The area of A is the number of its vertices. Typically, the graph in question is a regular lattice with the orientation of the bonds corresponding to some preferred direction. Examples are given in the next section. Directed animals are geometrical entities whose properties have been extensively studied over the past fifteen odd years due to their interest in both combinatorics and statistical physics. Few exact results are known. In 1982, Dhar, Phani and Barma gave two conjectures on the number of directed animals on the square and triangular lattices [1]. These conjectures can be restated in the form o...
A note on the enumeration of directed animals via gas considerations
 ANN. APPL. PROBAB
, 2009
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THE THEORY OF HEAPS AND THE CARTIER–FOATA MONOID
"... Abstract. We present Viennot’s theory of heaps of pieces, show that heaps are equivalent to elements in the partially commutative monoid of Cartier and Foata, and illustrate the main results of the theory by reproducing its application to the enumeration of parallelogram polyominoes due to Bousquet– ..."
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Cited by 2 (0 self)
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Abstract. We present Viennot’s theory of heaps of pieces, show that heaps are equivalent to elements in the partially commutative monoid of Cartier and Foata, and illustrate the main results of the theory by reproducing its application to the enumeration of parallelogram polyominoes due to Bousquet–Mélou and Viennot. 1.
Haruspicy 3: The anisotropic generating function of directed bondanimals is not Dfinite
 J. COMB. TH., SERIES A
, 2006
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