Polyominoes on twisted cylinders (original) (raw)

Counting Polyominoes on Twisted Cylinders

Discrete Mathematics & Theoretical Computer Science, 2005

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.

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.

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.

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...

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

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.

On the Generation of 2-Polyominoes

2018

The class of 2-polyominoes contains all polyominoes P such that for any integer i, the first i columns of P consist of at most 2 polyominoes. We provide a decomposition that allows us to exploit suitable discrete dynamical systems to define an algorithm for generating all 2-polyominoes of area n in constant amortized time and space O(n).

Signed polyomino tilings by n-in-line polyominoes and Groebner bases

arXiv (Cornell University), 2014

Conway and Lagarias observed that a triangular region T (m) in a hexagonal lattice admits a signed tiling by three-in-line polyominoes (tribones) if and only if m ∈ {9d − 1, 9d} d∈N. We apply the theory of Gröbner bases over integers to show that T (m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m ∈ {dn 2 − 1, dn 2 } d∈N. Explicit description of the Gröbner basis allows us to calculate the 'Gröbner discrete volume' of a lattice region by applying the division algorithm to its 'Newton polynomial'. Among immediate consequences is a description of the tile homology group of the n-in-line polyomino.

Green matrices associated with generalized linear polyominoes

Linear Algebra and its Applications, 2014

A Polyomino is an edge-connected union of cells in the planar square lattice. Here we consider generalized linear polyominoes; that is, the polyominoes supported by a n × 2 lattice. In this paper, we obtain the Green function and the Kirchhoff index of a generalized linear polyomino as a perturbation of a 2n-path by adding weighted edges between opposite vertices. This approach deeply links generalized linear polyomino Green functions with the inverse M-matrix problem, and especially with the so-called Green matrices.