Generating functions for straight polyomino tilings of narrow rectangles (original) (raw)
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Computing Generating Functions for Rectangular Domino Tilings
This paper reports a work in progress whose aim is to develop a computational framework towards the generation and analysis of rectangular domino tilings, that is, nonoverlapping coverings of a given rectangle by dominoes. A given rectangle is assumed to be of size n × m, n and m positive integers and all tessellating dominoes are identical rectangles of size 1 × 2. Our problem is to automatically obtain the generating function of the number of domino tilings the rectangle possess.
Counting Domino Tilings of Rectangles via Resultants
Advances in Applied Mathematics, 2001
The classical cosine formula for enumerating domino tilings of a rectangle, due to Kasteleyn, Temperley, and Fisher is proved using a combination of standard tools from combinatorics and algebra. For further details see [4].
A special tiling of the rectangle
2001
We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.
Combinatorially Regular Polyomino Tilings
Discrete & Computational Geometry, 2006
Let T be a regular tiling of R 2 which has the origin 0 as a vertex, and suppose that ϕ : R 2 → R 2 is a homeomorphism such that i) ϕ(0) = 0, ii) the image under ϕ of each tile of T is a union of tiles of T , and iii) the images under ϕ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertices of T such that Λ is a lattice and ϕ| Λ is a group homomorphism.
Combinatorial properties of double square tiles
Theoretical Computer Science, 2013
We study the combinatorial properties and the problem of generating exhaustively double square tiles, i.e. polyominoes yielding two distinct periodic tilings by translated copies such that every polyomino in the tiling is surrounded by exactly four copies. We show in particular that every prime double square tile may be obtained from the unit square by applying successively some invertible operators on double squares. As a consequence, we prove a conjecture of Provençal and Vuillon [17] stating that these polyominoes are invariant under rotation by angle π.
Generalizations of two statistics on linear tilings
We study generalizations of two well-known statistics on linear square-and-domino tilings by considering only those dominos the right half of which covers a multiple of k, where k is a fixed positive integer. Using the method of generating functions, we derive explicit expressions for the joint distribution polynomials of the two statistics with the statistic that records the number of squares in a tiling. In this way, we obtain two families of q-generalizations of the Fibonacci polynomials. When k=1, our formulas reduce to known results concerning previous statistics. Special attention is paid to the case k=2. As a byproduct of our analysis several combinatorial identities are obtained.
6 S ep 2 00 1 A special tiling of the rectangle
2001
We count tilings of a rectangle of integer sides m − 1 and n − 1 by a special set of tiles. The result is obtained from the study of the kernel of the adjacency matrix of an m × n rectangular subgraph in Z × Z.
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 TILINGS OF FINITE LATTICE REGIONS
We survey existing methods for counting two-cell tilings of finite planar lattice regions. We provide the only proof of the Pfaffian-squared theorem the author is aware of which is simultaneously modern, elementary, and fully correct. Detailed proofs of Kasteleyn's Rule and Theorem are provided. Theorems of Percus and Gessel-Viennot are proved in the standard fashions, hopefully with some advantage in clarity. Results of Kuperberg's are broken down and proved, again with significantly greater detail than we were able to find in a review of existing literature. We aim to provide a cohesive and compact, if bounded, examination of the structures and techniques in question.