Discrete mathematics 36 1981 191203 northholland publishing company counting polyoneinoes. This was the first algorithm that generated polyominoes without repetitions. In this paper we describe a generalization of redelmeiers algorithm for counting twodimensional rectangular polyominoes re81, which counts all the above. They always have connected interiors, but are allowed to have holes. A decomino, or 10omino, is a polyomino of order 10, that is, a polygon in the plane made of 10 equalsized squares connected edgetoedge. There are also twodimensional polyominoes that lie on a triangular or hexagonal lattice. That is, a d d polycube of size n is a connected set of n cells of a d dimensional hypercubic lattice, where connectivity is through d. Counting lattice animals in high dimensions figure 1. This involves the enumeration of the symmetry classes of parallelogram polyominoes under cyclic subgroups of 2. Polyomino, polyhex and polyiamond tiling introduction. Redelmeier 1981 computed the number of free and fixed polyominoes for, and mertens 1990 gives a simple computer program.
Tetrads and their counting baltic journal of modern computing. The basic idea is that we begin with a single square, and from there. In a previous work we described the generalization from two to higher dimensions of a polyomino counting algorithm of redelmeier d. Redelmeiers algorithm, which produced the entries in table 14. On the other hand we can say that a triangular system is a part of a triangular grid with vertices of degree 6, consisting of all edges and vertices of some closed broken linec without. In this paper we provide improved lower and upper bounds on the asymptotic growth constant of polyiamonds, proving that it is between 2. Guttmann, statistics of lattice animals polyominoes. In a previous work we described the generalization from two to higher dimensions of a polyominocounting algorithm of redelmeier d. Hugh redelmeier department of computer science, university of toronto, toronto, ontario, m5s i a 7 canada received 18 september 1979 revised 12 august 1980 a polyomino is a connected collection of squares on an unbounded chessboard. A tetromino is a geometric shape composed of four squares, connected orthogonally. Counting polycubes without the dimensionality curse counting polycubes without the dimensionality curse aleksandrowicz, gadi. In this paper we describe a generalization of redelmeiers algorithm for counting twodimensional rectangular polyominoes, which counts all the above types of polyominoes. In addition, we describe an efficient implementation of redelmeiers serial algorithm rede81 for counting three dimensional polyominoes termed polycubes.
Toronto, ontario, m5s i a7 canada received 18 september 1979 revised 12 august 1980 a polyomino is a connected collection of squares on an unbounded. The reader is referred to the original paper rede81 for the full details. It can be optimized so that it counts each polyomino only once, rather than n times. Polyominoesontwistedcylinders computational geometry. I am trying to understand one of the most fundamental algorithms in this area, although im aware. Counting ddimensional polycubes and nonrectangular planar. A tetromino is a geometric shape composed of four squares, connected orthogonally i.
This, like dominoes and pentominoes, is a particular type of polyomino. Redelmeiers algorithm for counting lattice animals request pdf. As klarner has shown, this will allow us to tighten the bounds for the asympototic growth constant of polyominoes. The statisticalphysics literature provides extensive enumeration data of polycubes 14, 24, 23, the most comprehensive being by luther and mertens 21, in particular, listing a 3n up to n19. In this paper we describe a generalization of redelmeiers algorithm for counting twodimensional rectangular polyominoes re81, which counts all the above types of polyominoes. A triangular system is a triangularpmino without any holes. Polyominoes are made by gluing together finitely many squares along their edges. Counting polycubes without the dimensionality curse core. A polyomino word derived from domino is a geometric plane figure made of the union of finitely many edgeconnected squares from the regular square lattice. The same can be said about tetrads made of polyominoes or another polyforms. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally a popular use of tetrominoes is in the video game tetris, which refers to them under the name tetrimino. Enumerating polyominoes is a huge subject, and of course the answers depend on whether you are interested in free, onesided, or fixed polyominoes.
There is no known formula yielding the number of distinct polyominoes of the given number of squares. The five free tetrominoes, top to bottom i, o, z, t, l, marked with light and dark squares. Hugh redelmeier department of computer science, uniwrsity of toronto, toronto, ontario, m5s ia7 canada received 18 september 1979 revised 12 august 1980 a polyomino is a connected collection of squares on an unbounded chessboard. Generating functions for connected embeddings in a lattice. Pdf the exact enumeration of most interesting combinatorial problems has exponential. Stack overflow for teams is a private, secure spot for you and your coworkers to find and share information. Formulae and growth rates of highdimensional polycubes ronnie barequet. For many applications it would seem desirable to retain more information than just the bare number of clusters. Counting ddimensional polycubes and nonrectangular planar polyominoes aleksandrowicz and barequet, 2009. Lunnon has defined a triangularpmino as an edgeconnected configuration ofp cells from the triangle plane grid with vertices of degree 6. A polyomino is a plane geometric figure formed by joining one or more equal squares edge to. Polyominoes have a long history, going back to the. In this paper we describe a generalization of redelmeier s algorithm for counting twodimensional rectangular polyominoes re81, which counts all the above types of polyominoes. The best known method in terms of running time for counting.
Redelmeiers algorithm for counting lattice animals. Formulae and growth rates of highdimensional polycubes. Earlier counts were given by lunnon 16 up to size 16, by sykes and glen 20 up to size 22,1 and by aleksandrowicz and barequet 3 extending redelmeiers polyominocounting algorithm 19 up to size 31. Redelmeier s algorithm for counting lattice animals center for geometric computing dept.
In proceedings of the international conference on computational science, part iii, lecture notes in computer science, 2659 melbourne, australia, and st. Polyominoes with minimum siteperimeter and full set achievement games. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally a popular use of tetrominoes is in the video game tetris, which refers to them as. Improved bounds on the growth constant of polyiamonds gill barequety mira shalah abstract a polyiamond is an edgeconnected set of cells on the triangular lattice. Redelmeier 4 presented an improved algorithm for the enumeration of poly. The sequence giving the number of free polyominoes of each order sloanes a000105, ball and coxeter 1987 is shown in the second column below, and that for fixed polyominoes in the third column sloanes a014559. Yet another attack redelmeier, 1981, presentation 2. Given a machine with enough processors, and 3 days, we could enumerate the asyet unknown number of polyominoes larger than size 24 based on redelmeiers results, 1981.
In this video we present redelmeiers algorithm for counting polyominoes, its generalization for counting animals on any lattice, and our implementation of a. When rotations and reflections are not considered to be distinct shapes, there are 4,655 different free decominoes the free decominoes comprise 195 with holes and 4,460 without holes. Redelmeiers algorithm is a procedure for connectedsubgraph counting, where the underlying graph is induced by the square lattice. Advanced seminar on polyominoes and polycubes, fall 201415. Generating and counting triangular systems springerlink. Counting ddimensional polycubes an d nonrectangular planar. Pri stetju prostih poliomin je treba po tvorjenju vsake nomine preveriti simetrije. Counting polycubes without the dimensionality curse. Archived from the original pdf of technical report version on 20061126.
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