## 数学代写|组合学代写Combinatorics代考|NWI-IBC016

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

## 数学代写|组合学代写Combinatorics代考|Generating Function of Permutations by Inversions

In Section 1.1, we looked at descents of permutations. That is, we studied instances in which an entry in a permutation was larger than the entry directly following it. A more “global” permutation statistic is that of inversions. This statistic will look for instances in which an entry of a permutation is smaller than some entry following it (not necessarily directly).

DEFINITION 2.1 Let $p=p_1 p_2 \cdots p_n$ be a permutation. We say that $\left(p_i, p_j\right)$ is an inversion of $p$ if $ip_j$.
Example 2.2
Permutation 31524 has four inversions, namely $(3,1),(3,2),(5,2)$, and $(5,4)$.
This line of research started as early as 1901 [254]. In this section, we survey some of the most interesting results in this area. The number of inversions of $p$ will be denoted by $i(p)$, though some authors prefer $i n v(p)$. It is clear that $0 \leq i(p) \leq\left(\begin{array}{c}n \ 2\end{array}\right)$ for all $n$-permutations, and that the two extreme values are attained by permutations $12 \cdots n$ and $n(n-1) \cdots 1$, respectively. It is relatively easy to find the generating function enumerating all permutations of length $n$ with respect to their number of inversions.

THEOREM 2.3
For all positive integers $n \geq 2$,
$$\sum_{p \in S_n} z^{i(p)}=I_n(z)=(1+z)\left(1+z+z^2\right) \cdots\left(1+z+z^2+\cdots+z^{n-1}\right) .$$

PROOF We prove the statement by induction on $n$. In fact, we prove that each of the $n$ ! expansion terms of the product $I_n(z)$ corresponds to exactly one permutation in $S_n$. Moreover, the expansion term $z^{a_1} z^{a_2} \cdots z^{a_{n-1}}$ will correspond to the unique permutation in which, for each $i \in[n]$, the entry $i+1$ precedes exactly $a_i$ entries that are smaller than itself.

If $n=2$, then there are two permutations to count, $p=12$ has no inversions, and $p^{\prime}=21$ has one inversion. So $\sum_{p \in S_2} z^{i(p)}=1+z$ as claimed. Furthermore, $p=12$ is represented by the expansion term 1 , and $p^{\prime}=21$ is represented by the expansion term $z$.

## 数学代写|组合学代写Combinatorics代考|Explicit Definition of Determinants

There are several undergraduate mathematics courses and textbooks that only give a recursive definition of the determinant of a square matrix. That is, $\operatorname{det}\left(\begin{array}{ll}a & b \ c & d\end{array}\right)$ is defined to be equal to $a d-b c$, and then the determinant of the $n \times n$ matrix $A=\left(a_{i j}\right)$ is defined to be
$$\operatorname{det} A=\sum_{j=1}^n(-1)^{j-1} a_{1 j} A_{1 j}$$
where $A_{1 j}$ is the $(n-1) \times(n-1)$ matrix obtained from $A$ by removing the first row and the $j$ th column.

If that is the only definition of determinants the reader has seen, he may find the following result interesting.
THEOREM 2.21
Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix. Then we have
$$\operatorname{det} A=\sum_{p \in S_n}(-1)^{i(p)} a_{1 p_1} a_{2 p_2} \cdots a_{n p_n} .$$
That is, $\operatorname{det} A$ is obtained by taking all $n$ ! possible $n$-tuples of entries so that there is exactly one of the $n$ entries in each row and each column, multiplying the elements of each such $n$-tuple together, finally taking a signed sum of these $n$ ! products, where the sign is determined by the parity of $i(p)$, and $p$ is the permutation determined by each chosen $n$-tuple.

In other words, the $n$-tuples correspond to all possible placements of $n$ rooks on an $n \times n$ chessboard so that no two of them hit each other.

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Generating Function of Permutations by Inversions

$$\sum_{p \in S_n} z^{i(p)}=I_n(z)=(1+z)\left(1+z+z^2\right) \cdots\left(1+z+z^2+\cdots+z^{n-1}\right) .$$

## 数学代写|组合学代写Combinatorics代考|Explicit Definition of Determinants

$$\operatorname{det} A=\sum_{j=1}^n(-1)^{j-1} a_{1 j} A_{1 j}$$

$$\operatorname{det} A=\sum_{p \in S_n}(-1)^{i(p)} a_{1 p_1} a_{2 p_2} \cdots a_{n p_n} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|组合学代写Combinatorics代考|MAT21018

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

## 数学代写|组合学代写Combinatorics代考|Alternating Runs and Alternating Subsequences

The length of the longest alternating subsequence of a permutation is closely connected to the number of alternating runs as shown by the following proposition.
PROPOSITION 1.53
Let $n \geq 2$. Then $a_k(n)=\frac{1}{2}(G(n, k-1)+G(n, k))$.
PROOF If an $n$-permutation $p$ has $i$ alternating runs and starts in a descent, then $\operatorname{as}(\mathrm{p})=\mathrm{i}+1$ as can be seen by considering the entries of $p$ that are peaks or valleys, as well as the first and last entry of $p$. It follows from the pigeon-hole principle that $p$ cannot contain a longer alternating subsequence. If $p$ starts in an ascent, then $\operatorname{as}(\mathrm{p})=\mathrm{i}$ by similar considerations.

Therefore, the $n$-permutations $p$ satisfying $\operatorname{as}(\mathrm{p})=\mathrm{k}$ are precisely the $n$ permutations with $k-1$ alternating runs starting in a descent and the $n$ permutations with $k$ alternating runs starting in an ascent.

Proposition 1.53 implies that if $T_n(z)=\sum_{p \in S_n} z^{\text {as(p) }}$, then
$$T_n(z)=\frac{1}{2}(1+z) G_n(z) .$$
So the polynomials $T_n(z)$ have real roots only, and $\lfloor n / 2\rfloor$ of their roots are equal to -1 .
The first few polynomials $T_n(z)$ are shown below.

1. $T_1(z)=z$,
2. $T_2(z)=z+z^2$,
3. $T_3(z)=z+3 z^2+2 z^3$,
4. $T_4(z)=z+7 z^2+11 z^3+5 z^4$,
5. $T_5(z)=z+15 z^2+43 z^3+45 z^4+16 z^5$,
6. $T_6(z)=z+31 z^2+148 z^3+268 z^4+211 z^5+61 z^6$,
7. $T_7(z)=z+63 z^2+480 z^3+1344 z^4+1767 z^5+1113 z^6+272 z^7$.

## 数学代写|组合学代写Combinatorics代考|Alternating Permutations

Sometimes an entire permutation is an alternating sequence, leading to the following definition.

DEFINITION $\mathbf{1 . 5 4}$ We say that the $n$-permutation $p$ is alternating if the longest alternating subsequence of $p$ is of length $n$. Similarly, we say that $p$ is reverse alternating if the longest reverse alternating subsequence of $p$ is of length $n$.

For instance, 312 and 5241736 are alternating permutations. Clearly, $p$ is alternating if and only if its complement, that is, the $n$-permutation whose $i$ th entry is $n+1-p_i$, is reverse alternating.

The number of alternating $n$-permutations is called an Euler number (not to be confused with the Eulerian numbers $A(n, k)$ ) and is denoted by $E_n$. The reader is invited to verify that $E_2=1, E_3=2, E_4=5$, and $E_5=16$. The Euler numbers have a very interesting exponential generating function. This is the content of the next theorem.
THEOREM 1.55
Set $E_0=1=E_1$. Then the equality
$$E(z)=\sum_{n \geq 0} E_n \frac{z^n}{n !}=\sec z+\tan z$$

holds.
PROOF Let $L(n+1) R$ be an alternating or reverse alternating permutation of length $n+1$. So $L$ is the string on the left of the maximal entry, and $R$ is the string on the right of the maximal entry. Then $R$ is reverse alternating, and so is $L^r$, that is, the reverse of $L$.
This observation leads to the recurrence relation
$$2 E_{n+1}=\sum_{k=0}^n\left(\begin{array}{l} n \ k \end{array}\right) E_k E_{n-k},$$
for $n \geq 1$. In terms of generating functions, this is equivalent to
$$2 E^{\prime}(z)=E^2(z)+1,$$
with $E(0)=1$.

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Alternating Runs and Alternating Subsequences

$$T_n(z)=\frac{1}{2}(1+z) G_n(z) .$$

$T_1(z)=z$，

$T_2(z)=z+z^2$，

$T_3(z)=z+3 z^2+2 z^3$，

$T_4(z)=z+7 z^2+11 z^3+5 z^4$，

$T_5(z)=z+15 z^2+43 z^3+45 z^4+16 z^5$，

$T_6(z)=z+31 z^2+148 z^3+268 z^4+211 z^5+61 z^6$，

$T_7(z)=z+63 z^2+480 z^3+1344 z^4+1767 z^5+1113 z^6+272 z^7$．

## 数学代写|组合学代写Combinatorics代考|Alternating Permutations

$$E(z)=\sum_{n \geq 0} E_n \frac{z^n}{n !}=\sec z+\tan z$$

hold住。

$$2 E_{n+1}=\sum_{k=0}^n\left(\begin{array}{l} n \ k \end{array}\right) E_k E_{n-k},$$

$$2 E^{\prime}(z)=E^2(z)+1,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|组合学代写Combinatorics代考|MATH4306

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

## 数学代写|组合学代写Combinatorics代考|Alternating Runs

Let us modify the notion of ascending runs that we discussed in the last section. Let $p=p_1 p_2 \cdots p_n$ be a permutation. We say that $p$ changes direction at position $i$ if either $p_{i-1}p_{i+1}$, or $p_{i-1}>p_i<p_{i+1}$. In other words, $p$ changes directions when $p_i$ is either a peak or a valley.

DEFINITION $\mathbf{1 . 3 7}$ We say that $p$ has $k$ alternating runs if there are $k-1$ indices $i$ so that $p$ changes direction at these positions.

For example, $p=3561247$ has 3 alternating runs as $p$ changes direction when $i=3$ and when $i=4$. A geometric way to represent a permutation and its alternating runs by diagram is shown in Figure 1.7. The alternating runs are the line segments (or edges) between two consecutive entries where $p$ changes direction. So a permutation has $k$ alternating runs if it can be represented by $k$ line segments so that the segments go “up” and “down” exactly when the entries of the permutation do.

The origins of this line of work go back to the nineteenth century. More recently, D. E. Knuth [230] has discussed the topic in connection to sorting and searching.

Let $G(n, k)$ denote the number of $n$-permutations having $k$ alternating runs. There are significant similarities between these numbers and the Eulerian numbers. For instance, for fixed $n$, both sequences have real zeros only, and both satisfy similar recurrence relations. However, the sequence of the $G(n, k)$ is not symmetric. On the other hand, almost half of all roots of the generating function $G_n(z)=\sum_{p \in S_n} z^{r(p)}=\sum_{k \geq 1} G(n, k) z^k$ are equal to -1 . Here $r(p)$ denotes the number of alternating runs of $p$.

## 数学代写|组合学代写Combinatorics代考|Definitions and a Recurrence Relation

The concept of alternating subsequences in permutations was introduced by Richard Stanley in [299]. In this section, we describe some of the major results of this recently developed subject, and we explain the very close connection between alternating subsequences and alternating runs.

DEFINITION 1.47 An alternating subsequence in a permutation $p=$ $p_1 p_2 \cdots p_n$ is a subsequence $p_{i_1} p_{i_2} \cdots p_{i_k}$ so that
$$p_{i_1}>p_{i_2}p_{i_4}<\cdots .$$

Similarly, a reverse alternating subsequence in $p$ is a subsequence $p_{j_1} p_{j_2} \cdots p_{j_k}$ so that
$$p_{j_1}p_{i_3}\cdots$$
The length of the longest alternating subsequence of $p$ is denoted by as(p). For instance, if $p=3416527$, then 3165,31657 , and 427 are examples of alternating subsequences of $p$.

Example 1.48
If $p=35714268$, then $\mathrm{as}(\mathrm{p})=5$. Indeed, 31426 is an alternating subsequence of $p$ of length 5. On the other hand, no alternating subsequence of $p$ can contain more than one of the first three entries of $p$, and no alternating subsequence can contain more than two of the last three entries of $p$. Therefore, no alternating subsequence of $p$ can be longer than five.

# 组合学代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|组合学代写Combinatorics代考|MATH233

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合学代写Combinatorics代考|Bipartite Multigraphs

Let $G=(V, E)$ be a multigraph. Then $G$ is called bipartite, provided that the vertex set $V$ may be partitioned into two subsets $X$ and $Y$ so that each edge of $G$ has one vertex in $X$ and one vertex in $Y$. A pair $X, Y$ with this property is called a bipartition of $G$ (and of its vertex set $V$ ). Two vertices in the same part of the bipartition are not adjacent. As we did in Chapter 9 for bipartite graphs, we usually picture a bipartite multigraph so that the vertices in $X$ are on the left (thus called left vertices) and the vertices in $Y$ are on the right (thus called right vertices). ${ }^{33}$ Note that a bipartite multigraph does not have any loops. A multigraph that is isomorphic to a bipartite multigraph is also bipartite.

Example. A bipartite multigraph with bipartition $X, Y$, where $X=$ ${a, b, c, d}$ and $Y={u, v, w}$, is shown in Figure 11.20.

Example. Consider the graph $G$ shown in Figure 11.21. Although it is not apparent from the drawing, $G$ is a bipartite graph. This is because we may also draw $G$ as in Figure 11.22, which reveals that $G$ has a bipartition $X={a, c, g, h, j, k}, Y={b, d, e, f, i}$.

The previous example demonstrates that a drawing of a bipartite graph or a listing of its edges may not directly reveal the bipartite property. A description of the edges of a graph may reveal a bipartition of its vertices.

## 数学代写|组合学代写Combinatorics代考|Trees

Suppose we want to build a connected graph of order $n$, using the smallest number of edges that we can “get away with.” 38 One simple method of construction is to select one vertex and join it by an edge to each of the other $n-1$ vertices. The result is a complete bipartite graph $K_{1, n-1}$, called a star. The star $K_{1, n-1}$ is connected and has $n-1$ edges. If we remove any edge from it, we obtain a disconnected graph with a vertex meeting no edges. Another simple method of construction is to join the $n$ vertices in a path. The resulting graph also is connected, has $n-1$ edges, and if we remove any edge, we obtain a disconnected graph. Can we construct a connected graph with $n$ vertices that has fewer than $n-1$ edges?

Suppose we have a connected graph $G$ of order $n$. Let’s think of putting in the edges of $G$ one by one. Thus, we start with $n$ vertices and no edges and hence with a graph with $n$ connected components. Each time we put in an edge we can decrease the number of connected components by, at most, 1: If the new edge joins 2 vertices that were already in the same component, then the number of components stays the same; if the new edge joins 2 vertices that were in different components, then those two components become one and all others are unaltered. Since we start with $n$ components, and an edge can decrease the number of components by at most 1 , we require at least $n-1$ edges in order to reduce the number of components to 1 , that is, in order to get a connected graph. So we have proved the next elementary result.

Theorem 11.5.1 A connected graph of order $n$ has at least $n-1$ edges. Moreover, for each positive integer $n$, there exist connected graphs with exactly $n-1$ edges. Removing any edge from a connected graph of order $n$ with exactly $n-1$ edges leaves a disconnected graph, and hence each edge is a bridge.

# 组合学代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|组合学代写Combinatorics代考|MATH233

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合学代写Combinatorics代考|Sums of Positive Terms

Suppose that
$$a_n=\sum_{k=0}^{L_n} t_{n, k}$$
where $t_{n, k}>0$ and $L_n \rightarrow \infty$. Imagine $n$ fixed and think of $t_{n, k}$ as a sequence in $k$; i.e., $t_{n, 0}, t_{n, 1}$, … Let $r_k(n)=t_{n, k+1} / t_{n, k}$, the ratio of consecutive terms. Usually we simply write $r_k$ for $r_k(n)$. In practice we usually have one of four situations
(a) Decreasing terms $\left(r_k \leq 1\right.$ for all $\left.k\right)$. We will study this.
(b) Increasing terms ( $r_k \geq 1$ for all $k$ ) Convert to (a) by writing the sequence backwards:
$$a_n=\sum_{k=0}^{L_n} t_{n, k}=\sum_{i=0}^{L_n} t_{n, L_n-i}=\sum_{k=0}^{L_n} s_{n, k},$$
(c) Increasing, then decreasing ( $r_k \leq 1$ for $k<K_n$ and $r_k \geq 1$ for $k \geq K_n$ ): split the sum at $K_n$. This gives one sum like (a) and one like (b):
$$a_n=\sum_{k=0}^{L_n} t_{n, k}=\sum_{k=0}^{K_n-1} t_{n, k}+\sum_{k=0}^{M_n} u_{n, k}$$

where $M_n=L_n-K_n$ and $u_{n, k}=t_{n, k+K_n}$.
(d) Decreasing, then increasing $\left(r_k \geq 1\right.$ for $k<K_n$ and $r_k \leq 1$ for $k \geq K_n$ : Split into two as done for (c).

Suppose we are dealing with (a), decreasing terms, and that $\lim _{n \rightarrow \infty} r_k(n)=r$ exists for each $k$ and does not depend on $k$. This may sound unusual, but it is quite common. If $r=1$, we will call the terms slowly decreasing. If $|r|<1$, we will call the terms rapidly decreasing. The two sums obtained from Case (c) are almost always slowly decreasing and asymptotically the same.

## 数学代写|组合学代写Combinatorics代考|Generating Functions

In order to study asymptotic estimates from generating functions, it is necessary to know something about singularities of functions. Essentially, a singularity is a point where the function misbehaves in some fashion. The singularities that are encountered in combinatorial problems are nearly all due to either

• attempting to take the logarithm of zero,
• attempting to raise zero to a power which is not a positive integer,
or both. For example, $-\ln (1-x)$ has a singularity at $x=1$. The power of zero requires a bit of explanation. It includes the obviously bad situation of attempting to divide by zero; however, it also includes things like attempting to take the square root of zero. For example, $\sqrt{1-4 x}$ has a singularity at $x=1 / 4$. To explain why a nonintegral power of zero is bad would take us too far afield. Suffice it to say that the fact that $A$ has two square roots everywhere except at $A=0$ is closely related to this problem.

The following is stated as a principle because we need to be more careful about the conditions in order to have a theorem. For combinatorial problems, you can expect the more technical conditions to be satisfied.

Principle 11.6 Nice singularities Let $a_n$ be a sequence whose terms are positive for all sufficiently large $n$. Suppose that $A(x)=\sum_n a_n x^n$ converges for some value of $x>0$. Suppose that $A(x)=f(x) g(x)+h(x)$ where

• $f(x)=(-\ln (1-x / r))^b(1-x / r)^c, c$ is not a positive integer and we do not have $b=c=0$;
• $A(x)$ does not have a singularity for $-r \leq x<r$;
• $\lim _{x \rightarrow r} g(x)$ exists and is nonzero (call it $L$ );
• $h(x)$ does not have a singularity at $x=r$.
Then it is usually true that
$$a_n \sim \begin{cases}\frac{L(\ln n)^b(1 / r)^n}{n^{c+1} \Gamma(-c)}, & \text { if } c \neq 0 ; \ \frac{b L(\ln n)^{b-1}(1 / r)^n}{n}, & \text { if } c=0 ;\end{cases}$$
where $\Gamma$ is the Gamma function which we describe below.

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Sums of Positive Terms

$$a_n=\sum_{k=0}^{L_n} t_{n, k}$$

(a)所有$\left.k\right)$的递减项$\left(r_k \leq 1\right.$。我们会研究这个的。
(b)增加项($r_k \geq 1$ for all $k$)通过向后写序列转换为(a):
$$a_n=\sum_{k=0}^{L_n} t_{n, k}=\sum_{i=0}^{L_n} t_{n, L_n-i}=\sum_{k=0}^{L_n} s_{n, k},$$
(c)先增加，再减少($k<K_n$为$r_k \leq 1$, $k \geq K_n$为$r_k \geq 1$):在$K_n$上分割金额。这给出了一个类似(a)和一个类似(b)的和:
$$a_n=\sum_{k=0}^{L_n} t_{n, k}=\sum_{k=0}^{K_n-1} t_{n, k}+\sum_{k=0}^{M_n} u_{n, k}$$

(d)先减少后增加$\left(r_k \geq 1\right.$ ($k<K_n$)， $r_k \leq 1$ ($k \geq K_n$):和(c)一样分成两部分。

## 数学代写|组合学代写Combinatorics代考|Generating Functions

$f(x)=(-\ln (1-x / r))^b(1-x / r)^c, c$ 不是正整数，我们没有$b=c=0$;

$A(x)$ 对于$-r \leq x<r$没有奇点;

$\lim _{x \rightarrow r} g(x)$ 存在且非零(调用$L$);

$h(x)$ 在$x=r$没有奇点。

$$a_n \sim \begin{cases}\frac{L(\ln n)^b(1 / r)^n}{n^{c+1} \Gamma(-c)}, & \text { if } c \neq 0 ; \ \frac{b L(\ln n)^{b-1}(1 / r)^n}{n}, & \text { if } c=0 ;\end{cases}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|组合学代写Combinatorics代考|MATH108

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合学代写Combinatorics代考|Symmetries and Pólya’s Theorem

In this section we will discuss a generalization of the Burnside Lemma. We will then consider an important special case of this generalization, namely Pólya’s Theorem. You should review the statement and proof of the Burnside Lemma (Theorem 4.5 (p. 112)).

Let $S$ be a set with a permutation group $G$. Recall that we say $x, y \in S$ are equivalent if $y=g(x)$ for some $g \in G$. (These equivalence classes are referred to as orbits of $G$ in $S$.) Suppose further that there is a function $W$ defined on $S$ such that $W$ is constant on equivalence classes. This means that if $x$ and $y$ are equivalent, then $W(y)=W(x)$. We can rephrase ” $W$ is constant on equivalence classes” as ” $W(g(x))=W(x)$ for all $g \in G$ and all $x \in S . “$

You may have noticed that $W$ is not completely specified because we haven’t defined its range. We don’t really care what the range is as long as addition of range elements and multiplication of them by rationals is possible. Thus the range might be the real numbers, polynomials with rational coefficients, or lots of other things.

Let $\mathcal{E}$ be the set of equivalence classes of $S$ with respect to the group $G$. (Don’t confuse $\mathcal{E}$ with our notation for exponential generating functions.) If $B \in \mathcal{E}$, define $W(B)=W(y)$, where $y$ is any element of $B$. This definition makes sense because $W$ is constant on equivalence classes.
Theorem 11.7 The Weighted Burnside Lemma With the above definitions,
$$\sum_{B \in \mathcal{E}} W(B)=\frac{1}{|G|} \sum_{g \in G} N(g),$$
where $N(g)$ is the sum of $W(x)$ over all $x \in S$ such that $g(x)=x$.

## 数学代写|组合学代写Combinatorics代考|Asymptotic Estimates

The area of asymptotics deals with obtaining estimates for functions for large values of the variables and, sometimes, for values near zero. Since the domain of the functions we’re concerned with is the positive integers, these functions can be thought of as sequences $a_1, a_2, \ldots$ Since this section uses the terminology introduced in Appendix B, you may want to review it at this time.

A solid mathematical treatment of asymptotics requires more background than we are willing to assume and developing the background would take too much time. Therefore, the material in this section is not rigorous. Instead, we present several principles which indicate what the result will almost certainly be in common combinatorial situations. The intent of this section is to give you a feeling for the subject, some direction for future study and some useful rules of thumb.

Before launching into specific tools and examples, we’d like to set the stage a bit since you are probably unfamiliar with asymptotic estimates. The lack of specific examples may make some of this introductory material a bit vague, so you may want to reread it after completing the various subsections.

Suppose we are interested in a sequence of numbers. We have four methods of providing asymptotic information about the numbers. Here they are, with examples:

• A combinatorial description: say $B_n$ is the number of partitions of an $n$-set;
• A recursion: $F_0=1, F_1=2$ and $F_n=F_{n-1}+F_{n-2}$ for $n \geq 2$;
• A formula: the number of involutions of an $n$-set is
$$\sum_{j=0}^n \frac{n !}{j ! 2^j(n-2 j) !} ;$$
the number of unlabeled full binary RP-trees with $n$ leaves is $\frac{1}{n}\left(\begin{array}{c}2 n-2 \ n-1\end{array}\right)$;

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Symmetries and Pólya’s Theorem

$$\sum_{B \in \mathcal{E}} W(B)=\frac{1}{|G|} \sum_{g \in G} N(g),$$

## 数学代写|组合学代写Combinatorics代考|Asymptotic Estimates

$$\sum_{j=0}^n \frac{n !}{j ! 2^j(n-2 j) !} ;$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|组合学代写Combinatorics代考|Derivatives, Averages and Probability

statistics-lab™ 为您的留学生涯保驾护航 在代写组合学Combinatorics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合学Combinatorics代写方面经验极为丰富，各种代写组合学Combinatorics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合学代写Combinatorics代考|Derivatives, Averages and Probability

The fact that $x A^{\prime}(x)=\sum n a_n x^n$ can be quite useful in obtaining information about averages. We’ll explain how this works and then look at some examples.

Let $\mathcal{A}n$ be a set of objects of size $n$; for example, some kind of $n$-long sequences or some kind of $n$-vertex trees. For each $n$, make $\mathcal{A}_n$ into a probability space using the uniform distribution: $$\operatorname{Pr}(\alpha)=\frac{1}{\left|\mathcal{A}_n\right|} \text { for all } \alpha \in \mathcal{A}_n$$ (Probability is discussed in Appendix C.) Suppose that for each $n$ we have a random variable $X_n$ on $\mathcal{A}_n$ that counts something; for example, the number of ones in a sequence or the number of leaves on a tree. The average value (average number of ones or average number of leaves) is then $\mathbf{E}\left(X_n\right)$. Now let’s look at this in generating function terms. Let $a{n, k}$ be the number of $\alpha \in \mathcal{A}n$ with $X_n(\alpha)=k$; for example, the number of $n$-long sequences with $k$ ones or the number of $n$-vertex trees with $k$ leaves. Let $A(x, y)$ be the generating function $\sum{n, k} a_{n, k} x^n y^k$. By the definition of expectation and simple algebra,
$$\mathbf{E}\left(X_n\right)=\sum_k k \operatorname{Pr}\left(X_n=k\right)=\sum_k k \frac{a_{n, k}}{\left|\mathcal{A}n\right|}=\frac{\sum_k k a{n, k}}{\left|\mathcal{A}n\right|}=\frac{\sum_k k a{n, k}}{\sum_k a_{n, k}}$$
Let’s look at the two sums in the last fraction.
Since $\left[x^n\right] A(x, y)=\sum_k a_{n, k} y^k, \quad \sum_k a_{n, k}=\left[x^n\right] A(x, 1)$.
Since $\left[x^n\right] \frac{\partial A(x, y)}{\partial y}=\sum_k k a_{n, k} y^{k-1}, \quad \sum_k k a_{n, k}=\left[x^n\right] A_y(x, 1)$,
where $A_y$ stands for $\partial A / \partial y$. Putting this all together,
$$\mathbf{E}\left(X_n\right)=\frac{\left[x^n\right] A_y(x, 1)}{\left[x^n\right] A(x, 1)}$$
We can use the same idea to compute variance. Recall that $\operatorname{var}\left(X_n\right)=\mathbf{E}\left(X_n^2\right)-\mathbf{E}\left(X_n\right)^2$. Since (10.21) tells us how to compute $\mathbf{E}\left(X_n\right)$, all we need is a formula for $\mathbf{E}\left(X_n^2\right)$. This is just like the previous derivation except we need factors of $k^2$ multiplying $a_{n, k}$. We can get this by differentiating twice:
$$\sum_k k^2 a_{n, k}=\left.\left[x^n\right] \frac{\partial\left(y A_y(x, y)\right)}{\partial y}\right|{y=1}=\left[x^n\right]\left(A{y y}(x, 1)+A_y(x, 1)\right)$$

## 数学代写|组合学代写Combinatorics代考|The Rules of Sum and Product

Before the 1960’s, combinatorial constructions and generating function equations were, at best, poorly integrated. A common route to a generating function was:

Obtain a combinatorial description of how to construct the structures of interest; e.g., the recursive description of unlabeled full binary $R P$-trees.

Translate the combinatorial description into equations relating elements of the sequence that enumerate the objects; e.g., $b_n=\sum_{k=1}^{n-1} b_k b_{n-k}$, for $n>1$ and $b_1=1$.

Introduce a generating function for the sequence and substitute the equations into the generating function. Apply algebraic manipulation.

The result is a relation for the generating function.
From the 1960 ‘s on, various people have developed methods for going directly from a combinatorial construction to a generating function expression, eliminating Steps 2 and 3. These methods often allow us to proceed from Step 1 directly to Step 4. The Rules of Sum and Product for generating functions are basic tools in this approach. We study them in this section.

So far we have been thinking of generating functions as being associated with a sequence of numbers $a_0, a_1, \ldots$ which usually happen to be counting something. It is often helpful to think more directly about what is being counted. For example, let $\mathcal{B}$ be the set of unlabeled full binary RP-trees. For $B \in \mathcal{B}$, let $w(B)$ be the number of leaves of $B$. Then $b_n$ is simply the number of $B \in \mathcal{B}$ with $w(B)=n$ and so
$$\sum_{B \in \mathcal{B}} x^{w(B)}=\sum_n b_n x^n=B(x) .$$

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Derivatives, Averages and Probability

$$\mathbf{E}\left(X_n\right)=\sum_k k \operatorname{Pr}\left(X_n=k\right)=\sum_k k \frac{a_{n, k}}{\left|\mathcal{A}n\right|}=\frac{\sum_k k a{n, k}}{\left|\mathcal{A}n\right|}=\frac{\sum_k k a{n, k}}{\sum_k a_{n, k}}$$

$$\mathbf{E}\left(X_n\right)=\frac{\left[x^n\right] A_y(x, 1)}{\left[x^n\right] A(x, 1)}$$

$$\sum_k k^2 a_{n, k}=\left.\left[x^n\right] \frac{\partial\left(y A_y(x, y)\right)}{\partial y}\right|{y=1}=\left[x^n\right]\left(A{y y}(x, 1)+A_y(x, 1)\right)$$

## 数学代写|组合学代写Combinatorics代考|The Rules of Sum and Product

$$\sum_{B \in \mathcal{B}} x^{w(B)}=\sum_n b_n x^n=B(x) .$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。