Combinatorics of diagonally convex directed polyominoes

Feretić, Svjetlan; Svrtan, Dragutin

doi:10.1016/s0012-365x(96)83012-3

Cited by 10 publications

(8 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These are themselves a subset of another class of compact clusters studied in the literature, named "fully directed compact" by physicists [46,47] and more precisely "diagonally convex directed" (DCD) by mathematicians [48,49]. DCD animals may be characterized as having the minimum accessible perimeter for a given length (i.e., unoccupied sites having an occupied predecessor).…”

Section: Numerical Resultsmentioning

confidence: 99%

On directed interacting animals and directed percolation

Knežević

Vannimenus

2002

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Abstract.We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n = 17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent φ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that φ is equal to the Fisher exponent σ governing the size distribution of large directed percolation clusters.

show abstract

Section: Numerical Resultsmentioning

confidence: 99%

On directed interacting animals and directed percolation

Knežević

Vannimenus

2002

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

show abstract

“…which is also the number of ternary trees with k internal nodes. Formula (2) was found by Delest and Fédou [5] in 1989, and equation (1) was found by Feretić and Svrtan [9] in 1994. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 86%

The perimeter generating function for nondirected diagonally convex polyominoes

Feretić¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes, researchers also gave a look at diagonally convex polyominoes (DCPs), but noticed an awkward feature: when the last diagonal of a DCP is deleted, the remaining object is not always a polyomino. So researchers focused their attention on directed DCPs. (A directed DCP is such a DCP that remains a polyomino when, for any i, its last i diagonals are deleted.) Directed DCPs gradually became well understood, whereas general DCPs have remained unexplored up to now.In this paper, we finally face general DCPs. Modulo a little trick, which saves us from dealing with non-polyominoes, we use the layered approach (described in chapter 3 of the book "Polygons, Polyominoes and Polycubes", edited by Anthony Guttmann). The computations are of remarkable bulk. Our main result is the perimeter generating function for DCPs; we denote it D(d, x). The function D(d, x) is algebraic and satisfies an equation of degree eight. The formula for D(d, x) is about eight pages long. That formula involves nine polynomials in d and x, and each of those polynomials is of degree 58 or more in x. The interested reader can view the formula for D(d, x) in the Maple worksheet attached to this paper.

show abstract

“…Polyominoes are widely used in physics and chemistry for modelling and they have long been studied by mathematicians and computer scientists (see [11,12,13] and the references given there). Concerning polyominoes with some lineconvexity properties some important results are already known.…”

Section: Diagonally Convex Directed Polyominoesmentioning

confidence: 99%

“…In [11] the authors studied the number of (horizontally and vertically) convex polyominoes reconstructible from their orthogonal projections and showed that in this class ambiguity can be very high. Moreover, in [12] a method is given to enumerate diagonally convex directed polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, etc.). Recently, in [14] the author stressed the importance of finding classes of polyominoes where the reconstruction from two projections can be solved uniquely in polynomial time.…”

Section: Diagonally Convex Directed Polyominoesmentioning

confidence: 99%

The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections

Balázs

2006

Discrete Geometry for Computer Imagery

View full text Add to dashboard Cite

Abstract. The number of line-convex directed polyominoes with given horizontal and vertical projections is studied. It is proven that diagonally convex directed polyominoes are uniquely determined by their orthogonal projections. The proof of this result is algorithmical. As a counterpart, we show that ambiguity can be exponential if antidiagonal convexity is assumed about the polyomino. Then, the results are generalised to polyominoes having convexity property along arbitrary lines.

show abstract

Combinatorics of diagonally convex directed polyominoes

Cited by 10 publications

References 12 publications

On directed interacting animals and directed percolation

On directed interacting animals and directed percolation

The perimeter generating function for nondirected diagonally convex polyominoes

The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections

Contact Info

Product

Resources

About