Related papers: Colored vertex models and $k$-tilings of the Aztec…
In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size of the…
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…
We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec…
We analyze domino tilings of the two-periodic Aztec diamond by means of matrix valued orthogonal polynomials that we obtain from a reformulation of the Aztec diamond as a non-intersecting path model with periodic transition matrices. In a…
The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three…
In this article we define a generalization of the domino shuffling algorithm for tilings of the Aztec diamond to the interacting $k$-tilings recently introduced by S. Corteel, A. Gitlin, and the first author. We describe the algorithm both…
The expanded Aztec diamond is a generalized version of the Aztec diamond, with an arbitrary number of long columns and long rows in the middle. In this paper, we count the number of domino tilings of the expanded Aztec diamond. The exact…
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…
We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the…
We give a bijective proof of the Aztec diamond theorem, stating that there are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order $n$. The proof in fact establishes a similar result for non-intersecting families of $n+1$ Schr\"oder…
In this paper, we give inductive sum formulas to calculate the number of diagonally symmetric, and diagonally \& anti-diagonally symmetric domino tilings of Aztec Diamonds. As a byproduct, we also find such a formula for the unrestricted…
We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the…
We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of…
The Aztec diamond of order $n$ is the union of lattice squares in the plane intersecting the square $|x|+|y|<n$. The Aztec diamond theorem states that the number of domino tilings of this shape is $2^{n(n+1)/2}$. It was first proved by…
We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic Combinatoric, 1992) by showing that the numbers of tilings of a certain family of regions…
We introduce a family of domino tilings that includes tilings of the Aztec diamond and pyramid partitions as special cases. These tilings live in a strip of $\mathbb{Z}^2$ of the form $1 \leq x-y \leq 2\ell$ for some integer $\ell \geq 1$,…
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…
Di Francesco introduced Aztec triangles as combinatorial objects for which their domino tilings are equinumerous with certain sets of configurations of the twenty-vertex model that are the main focus of his article. We generalize Di…
We use the subgraph replacement method to investigate new properties of the tilings of regions on the square lattice with diagonals drawn in. In particular, we show that the centrally symmetric tilings of a generalization of the Aztec…
We compute the algebraic equation for arctic curves of the Aztec diamond with a doubly (quasi-)periodic weight structure and obtain similar results for certain models of the hexagon. In particular, we determine the algebraic degree of such…