【读书笔记】格子路径计数LatticePathEnumeration 学一半的笔记

阅读原文时间：2023年07月09日阅读：3

流水账流水账这篇什么都不是

方法
10.2 Lattice paths without restrictions 无限制格子路径
- * 2维的例子，从(a,b)到(c,d)，允许(0,1)和(1,0)
10.3 Linear boundaries of slope 1
- * Theorem 10.3.1 在y=x对角线下
  - Theorem 10.3.3 在两条斜率1的对角线之间
10.4 Simple paths with linear boundaries of rational slope,部分一
- * Theorem 10.4.1 rational Catalan number
10.5 Simple paths with linear boundaries with rational slope,部分二
10.6 Simple paths with a piecewise linear boundary
- * Theorem 10.6.1
10.7 Simple paths with general boundaries
- * Theorem 10.7.1
10.8 Elementary results on Motzkin and Schröder paths
- * 一些著名paths
  - Theorem 10.8.1
  - Corollary 10.8.2
10.9 A continued fraction for the weighted counting of Motzkin paths
- * Theorem 10.9.1 Motzkin path带权重
  - Theorem 10.9.2 Motzkin和Schröder number的GF
  - Theorem 10.9.3 Dyck path带权重
10.10 Lattice paths and orthogonal polynomials
10.11 Motzkin paths in a strip
10.12 Further results for lattice paths in the plane
10.13 Non-intersecting lattice paths
- * 一些定义
  - Theorem 10.13.1
10.14 Lattice paths and their turns
10.15 Multidimensional lattice paths
10.16 Multidimensional lattice paths bounded by a hyperplane
10.17 Multidimensional paths with a general boundary
10.18 The reﬂection principle in full generality
10.19 q-Counting of lattice paths and Rogers–Ramanujan identities
10.20 Self-avoiding walks

吐个槽，以为逛组合学就不要看到大段复杂的公式了，结果。。。后面的哪些公式\(\omega\varpi\)都出来了，还一堆上下标。。。\(\sum\)下面求和范围都密密麻麻写两行。。。

方法

如果某人尝试去列出格子路径计数问题中的一些重要方法，那么会包括下面这些：

生成函数；拉格朗日反演公式；residue calcus
一一映射
reflection principle
cycle lemma
转移函数方法
核方法
the path switching involution for non-intersecting lattice paths
manipulation of two-rowed arrays for turn enumeration
正交多项式；连分数

10.2 Lattice paths without restrictions 无限制格子路径

2维的例子，从(a,b)到(c,d)，允许(0,1)和(1,0)

\[|L((a, b) \rightarrow(c, d))|=\left(\begin{array}{c}
c+d-a-b \\
c-a
\end{array}\right)
\]

n维的例子，从\(\mathbf{a}\)到\(\mathbf{e}\),每次允许一个维度+1

\[|L(\mathbf{a} \rightarrow \mathbf{e})|=\left(\begin{array}{c}
\sum_{i=1}^{d}\left(e_{i}-a_{i}\right) \\
e_{1}-a_{1}, e_{2}-a_{2}, \ldots, e_{d}-a_{d}
\end{array}\right):=\frac{\left(\sum_{i=1}^{d}\left(e_{i}-a_{i}\right)\right) !}{\left(e_{1}-a_{1}\right) !\left(e_{2}-a_{2}\right) ! \cdots\left(e_{d}-a_{d}\right) !}
\]

2维的例子，从(a,b)到(c,d)，允许\((0,\pm1)\)和\((\pm1,0)\),走n个steps

\[\left|L_{n}((a, b) \rightarrow(c, d) ;\{(\pm 1,0),(0,\pm 1)\})\right|=\left(\begin{array}{c}
n \\
\frac{n+c+d-a-b}{2}
\end{array}\right)\left(\begin{array}{c}
n \\
\frac{n+c-d-a+b}{2}
\end{array}\right)
\]

2维的例子，从(a,b)到(c,d)，允许(0,1)和(1,0)和(1,1)

\[|L((a, b) \rightarrow(c, d) ;{(1,0),(0,1),(1,1)}|=\sum_{k=0}^{c-a}\left(\begin{array}{c}
c+d-a-b-k \\
k, c-a-k, d-b-k
\end{array}\right)
\]

一个带权重计数，联系了格子路径计数和整数分拆

记号\(a(P)\)表示path \(P\)的水平step和x轴夹的面积的代数和

一个q-模拟

\[G F\left(L((a, b) \rightarrow(c, d)) ; q^{a(.)}\right)=q^{b(c-a)}\left[\begin{array}{c}
c+d-a-b \\
c-a
\end{array}\right]_{q}
\]

举例：从(0,0)到(2,2)有6种

是UURR，URUR，RURU，RRUU，URRU，RUUR

\[q^4+q^3+q^1+q^0+q^2+q^2=\left[\begin{array}{c}
4 \\
2
\end{array}\right]_{q}=\frac{\left(1-q^{4}\right)\left(1-q^{3}\right)}{(1-q)\left(1-q^{2}\right)}=\left(1+q^{2}\right)\left(1+q+q^{2}\right)=1+q+2 q^{2}+q^{3}+q^{4}
\]

10.3 Linear boundaries of slope 1

Theorem 10.3.1 在y=x对角线下

\[|L((a, b) \rightarrow(c, d) \mid x \geq y)|=\left(\begin{array}{c}
c+d-a-b \\
c-a
\end{array}\right)-\left(\begin{array}{c}
c+d-a-b \\
c-b+1
\end{array}\right)
\]

其中\(a \geq b，c\geq d\)

特例有：

ballot problem

\[|L((0,0) \rightarrow(c, d) \mid x \geq y)|=\frac{c+1-d}{c+d+1}\left(\begin{array}{c}
c+d+1 \\
d
\end{array}\right) \ where \ c\geq d
\]

卡特兰数

\[|L((0,0) \rightarrow(n, n) \mid x \geq y)|=\frac{1}{n+1}\left(\begin{array}{c}
2 n \\
n
\end{array}\right)
\]

Theorem 10.3.3 在两条斜率1的对角线之间

\[\begin{array}{l}
|L((a, b) \rightarrow(c, d) \mid x+t \geq y \geq x+s)| \\
\quad=\sum_{k \in \mathbb{Z}}\left(\left(\begin{array}{c}
c+d-a-b \\
c-a-k(t-s+2)
\end{array}\right)-\left(\begin{array}{c}
c+d-a-b \\
c-b-k(t-s+2)+t+1
\end{array}\right)\right)
\end{array}
\]

其中\(a+t\geq b\geq a+s\) and \(c+t\geq d\geq c+s\)

如果利用一些余数微积分的知识，我们可以改写成有sine和cosine的formula，这样容易分析渐进性质：

\[\begin{array}{c}
|L((a, b) \rightarrow(c, d) \mid x+t \geq y \geq x+s)| \\
=\sum\limits_{k=1}^{\lfloor(t-s+1) / 2\rfloor}\frac{4}{t-s+2}\left(2 \cos \frac{\pi k}{t-s+2}\right)^{c+d-a-b}
\cdot \sin \left(\frac{\pi k(a-b+t+1)}{t-s+2}\right) \cdot \sin \left(\frac{\pi k(c-d+t+1)}{t-s+2}\right)
\end{array}
\]

10.4 Simple paths with linear boundaries of rational slope,部分一

Theorem 10.4.1 rational Catalan number

\[|L((0,0) \rightarrow(r, s) \mid s x \geq r y)|=\frac{1}{r+s}\left(\begin{array}{c}
r+s \\
r,s
\end{array}\right)
\]

这里没错。。。我找了个pdf

这个数如今被称为有理卡特兰数，

如果让\(r=n,s=n+1\)，那么成为卡特兰数\(\frac{1}{n+1}\left(\begin{array}{c}
2n \\
n
\end{array}\right)\)

Theorem 10.4.5

\[|L((0,0) \rightarrow(c, d) \mid x \geq \mu y)|=\frac{c-\mu d+1}{c+d+1}\left(\begin{array}{c}
c+d+1 \\
d
\end{array}\right)
\]

其中，\(\mu\)是一个非负整数，\(c\geq \bold{\mu} d\)

Lemma 10.4.6 Cycle Lemma

Theorem 10.4.7

10.5 Simple paths with linear boundaries with rational slope,部分二

10.6 Simple paths with a piecewise linear boundary

picewise 分段的，逐段的

像下面这样

Theorem 10.6.1

10.7 Simple paths with general boundaries

Theorem 10.7.1

\[\left|L\left(\left(0, b_{1}\right) \rightarrow\left(n, a_{n}\right) \mid \mathbf{a} \geq \mathbf{y} \geq \mathbf{b}\right)\right|=\operatorname{det}_{1 \leq i, j \leq n}\left(\left(\begin{array}{c}
a_{i}-b_{j}+1 \\
j-i+1
\end{array}\right)\right)
\]

10.8 Elementary results on Motzkin and Schröder paths

一些著名paths

常见的就这么几种

Motzkin paths 允许(1,1),(1,-1),(1,0) 不会到x轴下方

Schröder paths 允许(1,1),(1,-1),(2,0) 不会到x轴下方

Catalan paths 允许(1,1),(1,-1) 不会到x轴下方

Dyck paths 上面的3种只要出发点和终点都在x轴上，就叫做Dyck paths

Theorem 10.8.1

Motzkin paths enumeration

\[\begin{array}{l}
L((a, b) \rightarrow(c, d) ; M \mid y \geq 0) \mid \\
\quad=\sum_{k=0}^{c-a}\left(\begin{array}{c}
c-a \\
k
\end{array}\right)\left(\left(\begin{array}{c}
c-a-k \\
(c+d-k-a-b) / 2
\end{array}\right)-\left(\begin{array}{c}
c-a-k \\
(c+d-k-a+b+2) / 2
\end{array}\right)\right)
\end{array}
\]

Schröder paths enumeration

\[\begin{array}{c}
|L((a, b) \rightarrow(c, d) ; S \mid y \geq 0)|=\sum_{k=0}^{(c-a) / 2}\left(\begin{array}{c}
c-a-k \\
k
\end{array}\right) \\
\cdot\left(\left(\begin{array}{c}
c-a-2 k \\
(c+d-2 k-a-b) / 2
\end{array}\right)-\left(\begin{array}{c}
c-a-2 k \\
(c+d-2 k-a+b+2) / 2
\end{array}\right)\right)
\end{array}
\]

Corollary 10.8.2

出发点和结束点都在x轴上的Motzkin paths计数

\[|L((0,0) \rightarrow(n, 0) ; M \mid y \geq 0)|=\sum_{k=0}^{\lfloor n / 2\rfloor}\left(\begin{array}{c}
n \\
2 k
\end{array}\right) \frac{1}{k+1}\left(\begin{array}{c}
2 k \\
k
\end{array}\right)
\]

出发点和结束点都在x轴上的Schröder paths计数

\[|L((0,0) \rightarrow(n, 0) ; S \mid y \geq 0)|=\sum_{k=0}^{n / 2}\left(\begin{array}{c}
n / 2+k \\
2 k
\end{array}\right) \frac{1}{k+1}\left(\begin{array}{c}
2 k \\
k
\end{array}\right)
\]