Enumeration of parallelogram polyominoes with given bond and site perimeter

Delest, Maylis; Gouyou-Beauchamps, Dominique; Vauquelin, Bernard

doi:10.1007/bf01788555

Cited by 25 publications

(17 citation statements)

References 11 publications

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“…The solutions to all three generating functions can be most simply expressed in terms of the solutions in equation (18), which are the seven first order solutions of L TIP 12 = LCLM(L T 8 , L I 8 , L P 8 ), plus solutions of M 6 = N 3 · N 2 · N 1 . We here focus on the solutions of the linear differential operator L I 8 , since the solutions of the other two generating functions are easily related to the solution of L I 8 by relationships (19) and (20).…”

Section: Resultsmentioning

confidence: 99%

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The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons

Assis¹,

Hoeij²,

Maillard³

2016

J. Phys. A: Math. Theor.

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We consider the isotropic perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons, whose 8th order linear Fuchsian ODEs are previously known. We derive simple relationships between the three generating functions, and show that all three generating functions are joint solutions of a common 12th order Fuchsian linear ODE. We find that the 8th order differential operators can each be rewritten as a direct sum of a direct product, with operators no larger than 3rd order. We give closed-form expressions for all the solutions of these operators in terms of 2 F 1 hypergeometric functions with rational and algebraic arguments. The solutions of these linear differential operators can in fact be expressed in terms of two modular forms, since these Self-avoiding walks (SAWs) and self-avoiding polygons (SAPs) have long been studied in enumerative combinatorics as models of percolation, polymers, surface roughness, and more [1], although both their generating functions remain unsolved to this day. Several classes of SAWs and SAPs have been solved by imposing either convexity or directedness constraints, or both. Within a class, walks and polygons usually havePerimeter g.f. of 3-choice, imperfect, 1-punctured staircase polygons 2 the same growth constant, also known as the connective constant, although recently prudent polygons have been shown to be exponentially sparse among prudent walks [2].The study of SAPs and its sub-categories involves the search for exact expressions of their generating functions as a function of various parameters of interest. These include the perimeter, width, height, site perimeter, left and right corners, and area, and for certain classes of SAPs, a generating function has been found which include all of these parameters explicitly, e.g. [3]. Among the known generating functions, rational, algebraic, D-finite, non D-finite, and natural boundaries have been derived (see [4] for a good review). Furthermore, among still unsolved classes, it is possible to prove results concerning the nature of the unsolved generating function. For example, the anisotropic perimeter generating function for the full SAPs class has been proven to not be a D-finite function in [5]. The wide variety of types of functions which arise in the study of SAPs offers an intriguing source of knowledge for what constitutes exact solutions in statistical mechanics.Among known perimeter generating functions are rational functions, algebraic functions, q-series, and natural boundaries [4], and quite generically the nature of the isotropic and anisotropic perimeter generating functions are the same. In the case of column-convex but not row-convex SAPs, the area generating functions are simpler than the corresponding perimeter generating function, being rational functions [4]. However, all known cases of area-perimeter generating functions involve q-series [4].Three-choice and 1-punctured staircase polygons are two classes of SAPs well studied in the literature [6,7], known to be D-finite functions...

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Section: Resultsmentioning

confidence: 99%

“…The site perimeter is a relevant quantity in percolation theory, and for staircase polygons it can be computed from the perimeter and the number of corners the polygon has. The perimeter-corner generating function is given in [19].…”

mentioning

confidence: 99%

The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons

Assis¹,

Hoeij²,

Maillard³

2016

J. Phys. A: Math. Theor.

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“…(20). The above result can be used to infer the limit distribution of area, along the lines of Section 3.2.…”

Section: Proposition 6 ([79]mentioning

confidence: 95%

“…In addition to perimeter, other counting parameters have been studied, such as horizontal and vertical perimeter, generalisations of area [79], radius of gyration [47,56], number of nearest-neighbour interactions [4], last column height [9] and site perimeter [20,7]. Also, motivated by applications in chemistry, symmetry subclasses of polygon models have been analysed [55,54,36].…”

Section: Polygon Modelsmentioning

confidence: 99%

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Limit Distributions and Scaling Functions

Richard

2009

Polygons, Polyominoes and Polycubes

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We discuss the asymptotic behaviour of models of lattice polygons, mainly on the square lattice. In particular, we focus on limiting area laws in the uniform perimeter ensemble where, for fixed perimeter, each polygon of a given area occurs with the same probability. We relate limit distributions to the scaling behaviour of the associated perimeter and area generating functions, thereby providing a geometric interpretation of scaling functions. To a major extent, this article is a pedagogic review of known results.

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