Exactly solved models of polyominoes and polygons
نویسندگان
چکیده
This chapter deals with the exact enumeration of certain classes of selfavoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of them is a recursive description of the polyominoes which then translates into a functional equation satisfied by the generating function. The first approach applies to classes of polyominoes having a linear recursive structure and results in a rational generating function. The second approach applies to classes of polyominoes having an algebraic recursive structure and results in an algebraic generating function. The third approach, commonly called the Temperley method, is based on the action of adding a new column to the polyominoes. We conclude by discussing some open questions.
منابع مشابه
0 Statistics of lattice animals ( polyominoes ) and polygons
We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, τ = 4.062570(8), for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound τ >...
متن کاملTwo generalizations of column-convex polygons
Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, . . . , p connected components. Then columnconvex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simpl...
متن کاملPunctured polygons and polyominoes on the square lattice
We use the finite lattice method to count the number of punctured staircase and selfavoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area generating functions. We show that the critical point is unchanged by a finite number of punctures, and that the critical exponent increases by a fixed amount for eac...
متن کاملInversion Relations, Reciprocity and Polyominoes
We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. Fo...
متن کاملPolygonal polyominoes on the square lattice
We study a proper subset of polyominoes, called polygonal polyominoes, which are defined to be self-avoiding polygons containing any number of holes, each of which is a self-avoiding polygon. The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no common vertices with each other, nor any common vertices with the surrounding polygon. There are no ‘holes-...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008