scholarly journals Domino tilings of the expanded Aztec diamond

2018 ◽  
Vol 341 (4) ◽  
pp. 1185-1191 ◽  
Author(s):  
Seungsang Oh
Keyword(s):  
Domino Tilings ◽  
Aztec Diamond

scholarly journals Symmetric matrices, Catalan paths, and correlations

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Emmanuel Tsukerman ◽  
Lauren Williams ◽  
Bernd Sturmfels
Keyword(s):  
Laurent Polynomial ◽  
Symmetric Matrices ◽  
Correlation Matrices ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
International Audience ◽  
Square Matrix

International audience Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.

scholarly journals A Bijection Proving the Aztec Diamond Theorem by Combing Lattice Paths

10.37236/2809 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Frédéric Bosio ◽  
Marc A. A. Van Leeuwen
Keyword(s):  
Lattice Paths ◽  
Square Grid ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Bijective Proof ◽  
Schröder Paths ◽  
Intersecting Families ◽  
Grid Points

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 $n+1$ Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an $n\times n$ square grid; these families are well known to be in bijection with tilings of the Aztec diamond. Our bijection is produced by an invertible "combing'' algorithm, operating on families of paths without non-intersection condition, but instead with the requirement that any vertical steps come at the end of a path, and which are clearly $2^{n(n+1)/2}$ in number; it transforms them into non-intersecting families.

scholarly journals A Simple Proof of the Aztec Diamond Theorem

10.37236/1915 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu
Keyword(s):  
Simple Proof ◽  
Lattice Paths ◽  
Hankel Determinants ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Schröder Numbers

Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schröder numbers.

scholarly journals A Generalization of Aztec Diamond Theorem, Part I

10.37236/3691 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Square Lattice ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
New Family ◽  
Schröder Paths

We consider a new family of 4-vertex regions with zigzag boundary on the square lattice with diagonals drawn in. By proving that the number of tilings of the new regions is given by a power 2, we generalize both Aztec diamond theorem and Douglas' theorem. The proof extends an idea of Eu and Fu for Aztec diamonds, by using a  bijection between domino tilings and non-intersecting Schröder paths, then applying Lindström-Gessel-Viennot methodology.

scholarly journals Correlations for the Novak process

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Eric Nordenstam ◽  
Benjamin Young
Keyword(s):  
Correlation Functions ◽  
Binomial Coefficient ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
The Matrix ◽  
International Audience ◽  
Lozenge Tilings

International audience We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam and Young (2011) and has many intriguing similarities with a more well-studied model, domino tilings of the Aztec diamond. The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j)-entry is the binomial coefficient $C(A, B_j-i)$ for indeterminate variables $A$ and $B_1, \dots , B_n.$ Nous étudions des pavages aléatoires d'une region dans le plan par des losanges qui s'appelle le demi-hexagone de Novak et nous calculons les corrélations de ce processus. Ce modèle a été introduit par Nordenstam et Young (2011) et a plusieurs similarités des pavages aléatoires d'un diamant aztèque par des dominos. La partie la plus difficile de cet article est le calcul de l'inverse d'une matrice ou l’élément (i,j) est le coefficient binomial $C(B_j-i, A)$ pour des variables $A$ et $B_1, \dots , B_n$ indéterminés.

scholarly journals Domino tilings of rectangles with fixed width

1980 ◽  
Vol 32 (1) ◽  
pp. 45-52 ◽  
Author(s):  
David Klarner ◽  
Jordan Pollack
Keyword(s):  
Domino Tilings

scholarly journals Asymptotic domino statistics in the Aztec diamond

2015 ◽  
Vol 25 (3) ◽  
pp. 1232-1278 ◽  
Author(s):  
Sunil Chhita ◽  
Kurt Johansson ◽  
Benjamin Young
Keyword(s):  
Aztec Diamond

scholarly journals A Note on Domino Shuffling

10.37236/1056 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
É. Janvresse ◽  
T. de la Rue ◽  
Y. Velenik
Keyword(s):  
Large Class ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Sufficient Condition ◽  
Aztec Diamond

We present a variation of James Propp's generalized domino shuffling, which provides an efficient way to obtain perfect matchings of weighted Aztec diamonds. Our modification is specially tailored to deal with cases when some of the weights are zero. This allows us to tile efficiently a large class of planar graphs, by embedding them in a large enough Aztec diamond. We also give a sufficient condition on the size of the latter diamond for the algorithm to succeed.

scholarly journals A Gessel–Viennot-Type Method for Cycle Systems in a Directed Graph

10.37236/1063 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Christopher R. H. Hanusa
Keyword(s):  
Directed Graph ◽  
Type Method ◽  
Domino Tilings ◽  
Cycle Systems ◽  
Insight Into

We introduce a new determinantal method to count cycle systems in a directed graph that generalizes Gessel and Viennot's determinantal method on path systems. The method gives new insight into the enumeration of domino tilings of Aztec diamonds, Aztec pillows, and related regions.

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