多米諾骨牌是一個二格骨牌，由兩個全等的正方形以邊相連組成。多米諾骨牌拼圖是在歐幾里德平面的格子區域中，由多個不重疊的多米諾骨牌鑲嵌而成，這裡我們以M(R)來表示在區域 中的多米諾骨牌拼圖數量。將格子區域中的每個正方形視為一個點、兩個相鄰的正方形構成一個邊，令為G(R)，那多米諾骨牌拼圖能對應到G(R)的完美匹配。

1961年，Temperley 和 Fisher [13] 給出一個公式來計算m列n行的矩形區域之拼圖方法數。在一個區域的拼圖方法數會依不同區域形狀而影響。有很多特殊區域可以對應到一些特別的數列。在本文中，我們將討論一些特殊區域的多米諾骨牌拼圖與非交錯路徑之間的對應關係，同時也給出了的多米諾骨牌拼圖方法數。

在第二章中，我們將介紹一些特殊區域中的多米諾骨牌拼圖。例如2列n行矩形區域的拼圖數量可以對應於斐波納契數列[7]。D. E. Knuth找到環狀方形區域的拼圖數量[6]。此外，Aztec Diamond區域 、Aztec Triangle區域的拼圖方法數會依序對應到不交錯的Schroder數列[9]以及Aztec Diamond的延伸區域之拼圖方法數會對應Delannoy數列[10]。

在第三章中，我們重新歸納Aztec Diamond和Schroder路徑之間對應[5]。並且證明在一般情況下，任何區域的多米諾骨牌拼圖和非交叉的路徑之間的一對一映成關係。

A domino is a polyomino of order 2 which is a polygon made of two equal-sized
squares connected edge-to-edge. A domino tiling of a region R in the
Euclidean plane is a tessellation of the region by non-overlapping dominoes,
and the number of ways of the domino tiling in the R region is denoted by
M(R). Equivalently, it is a perfect matching in the dual graph formed by
placing a vertex at the center of each square of the region and connecting
two vertices when they correspond to adjacent squares.
In 1961, Temperley and Fisher [11] gave a formula to count the number
of ways to cover an m × n rectangle. The number of the domino tilings of a
region is very sensitive to the shape of the region. There are special results
generated by opponent regions. In the thesis, we will give a viewpoint of
bijection between domino tiling and nonintersecting lattice paths as well as
give the formula for those special regions.
In Chapter 2, we will introduce the results having been proved of domino
tiling with simple regions, for instances: the region Fn which corresponds to
Fibonacci number in 2009 [6], the square ring Fm,n developed by Donald E.
Knuth in 2011 [5], the Aztec Diamond ADn, the agumented Aztec Diamond
Dn corresponding to Delannoy number [10] and the region Sn corresponding
to Schr¨oder number [2].
In Chapter 3, we reprove a bijection between the Aztec diamond and
disjoint Schr¨oder paths. Within giving a generalized bijection, we can find
the bijection between the nonintersecting lattice paths and domino tiling in
any region R.

致謝 i
摘要 ii
Abstract iii
Contents iv
List of Figures vi
1 Introduction 1
1.1 Basic notations 1
1.2 The main results of the thesis 3
2 The known results 5
2.1 The region 2 × n with Fibonacci numbers 5
2.2 The square Ring Fm,n with Fibonacci numbers 6
2.3 The Aztec Diamond 9
2.4 The augmented Aztec Diamond 9
2.5 The region Sn with the Schr¨oder numbers 10
3 Necessary and sufficient conditions 12
3.1 The bijection with the lattice paths 12
3.2 General proof 16
Bibliography 25

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