A New Simple Proof of the Aztec Diamond Theorem

The Electronic Journal of Combinatorics - Electr. J. Comb., 1998

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings.

Journal of Combinatorial Theory, Series A, 1999

In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing ``barriers'' in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of constrained tilings is independent of the placement of the barriers. We do not know of a simple combinatorial explanation of this fact; our proof uses the Jacobi Trudi identity. 1999 Academic Press An Aztec diamond of order n is a region composed of 2n(n+1) unit squares, arranged in bilaterally symmetric fashion as a stack of 2n rows of squares, the rows having lengths 2, 4, 6, ..., 2n&2, 2n, 2n, 2n&2, ..., 6, 4, 2. A domino is a 1-by-2 (or 2-by-1) rectangle. It was shown in [1] that the Aztec diamond of order n can be tiled by dominoes in exactly 2 n(n+1)Â2 ways. Here we study barriers, indicated by darkened edges of the square grid associated with an Aztec diamond. These are edges that no domino is permitted to cross. (If one prefers to think of a domino tiling of a region as a perfect matching of a dual graph whose vertices correspond to gridsquares and whose edges correspond to pairs of grid-squares having a shared edge, then putting down a barrier in the tiling is tantamount to removing an edge from the dual graph.) Fig. (left) shows an Aztec diamond of order 8 with barriers, and Fig. (right) shows a domino-tiling that is compatible with this placement of barriers.

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.

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.

The Ramanujan Journal, 2019

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a 1 × ∞ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a q-series identity. Using this method, they recover quite a few classical q-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Euler's Pentagonal Number Theorem along with an infinite family of generalizations.

Eprint Arxiv 1206 2230, 2012

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T , overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N-tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T ; or given N , determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is the second of four papers on this subject. In [1], we dealt with the case when ABC is similar to T , and the case when T is a right triangle. In this paper, we assume that ABC is not similar to T , and T is not a right triangle, and furthermore that if T has a 120 • angle then T is isosceles. The main theorem is that under those hypotheses, the only N-tilings are of an equilateral triangle by an isosceles tile with base angles π/6, and N three times a square. Under those hypotheses, there are only two families of tilings. There are tilings of an equilateral triangle ABC by an isosceles tile with base angles π/6; and there are newly-discovered "triquadratic tilings", which are treated in the third paper [2] in this series. These arise in the special case that 3α + 2β = π or 3β + 2α = π with α not a rational multiple of π, where α and β are the two smallest angles of the tile, and the sides of the tile have rational ratios. The case when the tile has an angle 2π/3 and is not isosceles is taken up in [3]. At the time of writing, our analysis of that case is incomplete, but there are no known tilings of any ABC in that case. We use techniques from linear algebra and elementary field theory, and in one case we use some algebraic number theory. We use some counting arguments and some elementary geometry and trigonometry.

Arxiv preprint math/9801068

In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/ √ 2 for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.

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].

2012

An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This paper takes up the case when $3\alpha + 2\beta = \pi$. Then there are (as was already known) exactly five possible shapes of $ABC$: either $ABC$ is isosceles with base angles $\alpha$, $\beta$, or $\alpha+\beta$, or the angles of $ABC$ are $(2\alpha,\beta,\alpha+\beta)$, or the angles of $ABC$ are $(2\alpha, \alpha, 2\beta)$. In each of these cases, we have discovered, and here exhibit, a family of previously unknown tilings. These are tilings that, as far as we know, have never been seen before. We also discovered, in each of the cases, a Diophantine equation involving $N$ and the (necessarily rational) number $s = a/c$ that has solutions if there is a tiling using tile $T$ of some $ABC$ not similar to $T$. By means of these Diophantine equations, some conclusions about the possible values of $N$ are drawn; in particular there are no tilings possible for values of $N$ of certain forms. We prove, for example, that there is no $N$-tiling with $N$ prime when $3\alpha + 2\beta = \pi$. These equations also imply that for each $N$, there is a finite set of possibilities for the tile $(a,b,c)$ and the triangle $ABC$. (Usually, but not always, there is just one possible tile.) These equations provide necessary, and in three of the five cases sufficient, conditions for the existence of $N$-tilings.

Journal of Mathematics and Music, 2009

A finite subset A of integers tiles the discrete line Z if the integers can be written as a disjoint union of translates of A. In some cases, necessary and sufficient conditions for A to tile the integers are known. We extend this result to a large class of nonperiodic tilings and give a new formulation of the Coven-Meyerowitz reciprocity conjecture which is equivalent to the Flugede conjecture in one dimension.