Aztec diamond

In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n. Here n is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are half-integers.^[1]

The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order n is 2^n(n+1)/2.^[2] The arctic circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.^[3]

An Aztec diamond of order 4, with 1024 domino tilings
One possible tiling.
A lozenge tiling of a hexagon chosen uniformly at random, with the "frozen" tiles being depicted in white. (Arctic Circle theorem.)

It is common to color the tiles in the following fashion. First consider a checkerboard coloring of the diamond. Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square, is colored in one color, and the other vertical tiles in a second. Similarly for horizontal tiles.

Non-Intersecting Paths

Something that is very useful for counting tilings is looking at the non-intersecting paths through its corresponding directed graph. If we define our movements through a tiling (domino tiling) to be

(1,1) when we are the bottom of a vertical tiling
(1,0) where we are the end of a horizontal tiling
(1,-1) when we are at the top of a vertical tiling

Then through any tiling we can have these paths from our sources to our sinks. These movements are similar to Schröder paths. For example, consider an Aztec Diamond of order 2, and after drawing its directed graph we can label its sources and sinks. It sources should be $a_{1},a_{2}$ are our sources and $b_{1},b_{2}$ are our sinks. On its directed graph, we can draw a path from $a_{i}$ to $b_{j}$ , this gives us a path matrix, $P_{2}$ ,

$P_{2}={\begin{bmatrix}a_{1}b_{1}&a_{2}b_{1}\\a_{1}b_{2}&a_{2}b_{2}\\\end{bmatrix}}={\begin{bmatrix}6&2\\2&2\\\end{bmatrix}}$

where $a_{i}b_{j}=$ all the paths from $a_{i}$ to $b_{j}$ . The number of tilings for order 2 is

det $(P_{2})=12-4=8=2^{2(2+1)/2}$

According to Lindstrom-Gessel-Viennot, if we let S be the set of all our sources and T be the set of all our sinks of our directed graph then

det $(P_{n})=$ number of non-intersecting paths from S to T.^[4]

Considering the directed graph of the Aztec Diamond, it has also been shown by Eu and Fu that Schröder paths and the tilings of the Aztec diamond are in bijection.^[5] Hence, finding the determinant of the path matrix, $P_{n}$ , will give us the number of tilings for the Aztec Diamond of order n.

Another way to determine the amount of tilings of the Aztec Diamond is using Hankel matrices of large and small Schröder numbers,^[5] using the method from Lindstrom-Gessel-Viennot again.^[4] Finding the determinant of these matrices gives us the number of number of non-intersecting paths of small and large Schröder numbers, which is in bijection with the tilings. Note: the small Schröder numbers are $\{1,1,3,11,45,\cdots \}=\{y_{0},y_{1},y_{2},y_{3},y_{4},\cdots \}$ and the large Schröder numbers are $\{1,2,6,22,90,\cdots \}=\{x_{0},x_{1},x_{2},x_{3},x_{4},\cdots \}$ , and in general our two Hankel matrices will be

$H_{n}={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\\x_{2}&x_{3}&\cdots &x_{n+1}\\\vdots &\vdots &&\vdots \\x_{n}&x_{n+1}&\cdots &x_{2n-1}\\\end{bmatrix}}$ and $G_{n}={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y_{2}&y_{3}&\cdots &y_{n+1}\\\vdots &\vdots &&\vdots \\y_{n}&y_{n+1}&\cdots &y_{2n-1}\\\end{bmatrix}}$

where det $(H_{n})=2^{2(n+1)/2}$ and det $(G_{n})=2^{2(n-1)/2}$ where $n\geq 1$ (It also true that det $(H_{n}^{0})=2^{2(n-1)/2}$ where this is the Hankel matrix like $H_{n}$ , but you start with $x_{0}$ instead of $x_{1}$ for the first entry of the matrix in the top left corner).

Sources

There are many tools to used throughout tiling projects, but two useful ones are GeoGebra and program created by Jim Propp, Greg Kuperburg, and David Wilson in SageMath to count the tilings of a shape.^[7] The link to this specific program is located in external links, under Tiling Program in Sage.

External links

References

↑ Stanley, Richard P. (1999), Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282
↑ Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James (1992), "Alternating-sign matrices and domino tilings. I", Journal of Algebraic Combinatorics. An International Journal, 1 (2): 111–132, doi:10.1023/A:1022420103267, ISSN 0925-9899, MR 1226347
↑ Jockusch, William; Propp, James; Shor, Peter (1998), Random Domino Tilings and the Arctic Circle Theorem, arXiv:math/9801068, Bibcode:1998math......1068J
1 2 Majumdar, Diptapriyo. "Advance Graph Algorithms: Lemma of Gessel Viennot" (PDF). Retrieved 22 April 2014.
1 2 Eu, Sen-Peng; Fu, Tung-Shan. "A Simple Proof of the Aztec Diamond". The Electroninc Journal of Combinatorics. Retrieved 20 April 2005.
1 2 Martinez, Megan; Kanoff, Ilene. "Domino Tiling and the Fibonacci numbers" (PDF). Retrieved 2 March 2018.
↑ Propp, Jim. "Cellular Automata/ TIlings". Jim Propp. Retrieved 3 March 2018.

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[1] Stanley, Richard P. (1999), Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282

[2] Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James (1992), "Alternating-sign matrices and domino tilings. I", Journal of Algebraic Combinatorics. An International Journal, 1 (2): 111–132, doi:10.1023/A:1022420103267, ISSN 0925-9899, MR 1226347

[3] Jockusch, William; Propp, James; Shor, Peter (1998), Random Domino Tilings and the Arctic Circle Theorem, arXiv:math/9801068, Bibcode:1998math......1068J

[Gessel-4] 1 2 Majumdar, Diptapriyo. "Advance Graph Algorithms: Lemma of Gessel Viennot" (PDF). Retrieved 22 April 2014.

[Eu-5] 1 2 Eu, Sen-Peng; Fu, Tung-Shan. "A Simple Proof of the Aztec Diamond". The Electroninc Journal of Combinatorics. Retrieved 20 April 2005.

[Fibonacci-6] 1 2 Martinez, Megan; Kanoff, Ilene. "Domino Tiling and the Fibonacci numbers" (PDF). Retrieved 2 March 2018.

[Propp-7] Propp, Jim. "Cellular Automata/ TIlings". Jim Propp. Retrieved 3 March 2018.

Aztec diamond

Non-Intersecting Paths

Other Tiling Problems

Sources

External links

References