Counting Polyominoes on Twisted Cylinders (original) (raw)

Polyominoes on twisted cylinders

Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13, 2013

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called twisted cylinder by the transfer matrix method. A bijective representation of the "states" of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method. † Supported by the Deutsche Forschungsgemeinschaft within the European Graduate Program Combinatorics, Geometry and Computation (No. GRK 588/2).

Counting polyominoes: Yet another attack

Discrete Mathematics, 1981

A polyomino is a connected collection of squares on an unbounded chessboard. There is no known formula yielding the number of distinct polyominoes of a given number of squares A polyomino enumeration method, faster than any previous, is presented. This method includes the calculation of the number of symmetric polyominoes. AU polyominoes containing up to 24 squares have been enumerated (using ten months of computer time). Previously, only polyominoes up to size 18 were enumerated.

Counting d-Dimensional Polycubes and Nonrectangular Planar Polyominoes

Springer eBooks, 2006

A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. Similarly, a d-dimensional polycube (for d ≥ 2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d − 1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. 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. For example, our program computed the number of distinct 3-D polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

2020

A ddd-dimensional polycube is a facet-connected set of cells (cubes) on the ddd-dimensional cubical lattice mathbbZd\mathbb{Z}^dmathbbZd. Let Ad(n)A_d(n)Ad(n) denote the number of ddd-dimensional polycubes (distinct up to translations) with nnn cubes, and lambdad\lambda_dlambdad denote the limit of the ratio Ad(n+1)/Ad(n)A_d(n{+}1)/A_d(n)Ad(n+1)/Ad(n) as ntoinftyn \to \inftyntoinfty. The exact value of lambdad\lambda_dlambdad is still unknown rigorously for any dimension dgeq2d \geq 2dgeq2; the asymptotics of lambdad\lambda_dlambdad, as dtoinftyd \to \inftydtoinfty, also remained elusive as of today. In this paper, we revisit and extend the approach presented by Klarner and Rivest in 1973 to bound A2(n)A_2(n)A2(n) from above. Our contributions are: Using available computing power, we prove that lambda2leq4.5252\lambda_2 \leq 4.5252lambda2leq4.5252. This is the first improvement of the upper bound on lambda2\lambda_2lambda2 in almost half a century; We prove that lambdadleq(2d−2)e+o(1)\lambda_d \leq (2d-2)e+o(1)lambdadleq(2d−2)e+o(1) for any value of dgeq2d \geq 2dgeq2, using a novel construction of a rational generating function which dominates that of the sequence left(Ad(n)right)\left(A_d(n)\right)left(Ad(n)right); For $d=3...

Polygonal polyominoes on the square lattice

Journal of Physics A: Mathematical and General, 2001

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-within-holes'. We use the finite-lattice method to count the number of polygonal polyominoes on the square lattice. Series have been derived for both the perimeter and area generating functions. It is known that while the critical point is unchanged by a finite number of holes, when the number of holes is unrestricted the critical point changes. The area generating function coefficients grow exponentially, with a growth constant greater than that for polygons with a finite number of holes, but less than that of polyominoes. We provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes. Also, we prove that, enumerating by perimeter, the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds.

Counting Polyominoes: A Parallel Implementation for Cluster Computing

Lecture Notes in Computer Science, 2003

The exact enumeration of most interesting combinatorial problems has exponential computational complexity. The finite-lattice method reduces this complexity for most two-dimensional problems. The basic idea is to enumerate the problem on small finite lattices using a transfer-matrix formalism. Systematically grow the size of the lattices and combine the results in order to obtain the desired series for the infinite lattice limit. We have developed a parallel algorithm for the enumeration of polyominoes, which are connected sets of lattice cells joined at an edge. The algorithm implements the finite-lattice method and associated transfer-matrix calculations in a very efficient parallel setup. Test runs of the algorithm on a HP server cluster indicates that in this environment the algorithm scales perfectly from 2 to 64 processors.

Universality and asymptotics of graph counting problems in non-orientable surfaces

Journal of Combinatorial Theory, 2010

Bender-Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants tg and pg for the orientable and the non-orientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series solution u(z) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of tg to all orders in 1/g for large g. The paper introduces a formal power series solution v(z) of a Riccati equation, gives a nonlinear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence pg and v(z). Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2-dimensional projective plane. Our conjecture implies analyticity of the O(N ) and Sp(N )-types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann-Hilbert approach, and provide ample numerical evidence for our results.

The Polygonal Cylinder and Its Hosoya Polynomial

2020

We introduce a polygonal cylinder Cm,n, using the Cartesian product of paths Pm and Pn and using topological identification of vertices and edges of two opposite sides of Pm × Pn, and give its Hosoya polynomial, which, depending on odd and even m, is covered in seven separate cases. Subject Classification (2010). 05C12; 05C30; 05C31

On the complexity of Jensen's algorithm for counting fixed polyominoes

Journal of Discrete Algorithms, 2007

Recently I. Jensen published a novel transfer-matrix algorithm for computing the number of polyominoes in a rectangular lattice. However, his estimation of the computational complexity of the algorithm (O((√ 2) n), where n is the size of the polyominoes), was based only on empirical evidence. In contrast, our research provides some solid proof. Our result is based primarily on an analysis of the number of some class of strings that plays a significant role in the algorithm. It turns out that this number is closely related to Motzkin numbers. We provide a rigorous computation that roughly confirms Jensen's estimation. We obtain the bound O(n 5/2 (√ 3) n) on the running time of the algorithm, while the actual number of polyominoes is about C4.06 n /n, for some constant C > 0.

Punctured polygons and polyominoes on the square lattice

Journal of Physics A: Mathematical and General, 2000

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 each puncture. The increase is 1.5 per puncture when enumerating by perimeter and 1.0 when enumerating by area. A refined estimate of the connective constant for polygons by area is given. A similar set of results is obtained for finitely punctured polyominoes. The exponent increase is proved to be 1.0 per puncture for polyominoes.