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## 数学代写|组合学代写Combinatorics代考|Exploiting generating functions and counting sequence

Exploiting generating functions and counting sequences. In this book we are going to see altogether more than a hundred applications of the symbolic method. Before engaging in technical developments, it is worth inserting a few comments on the way generating functions and counting sequences can be put to good use in order to solve combinatorial problems.

Explicit enumeration formuld. In a number of situations, generating functions are explicit and can be expanded in such a way that explicit formulae result for their coefficients. A prime example is the counting of general trees and of triangulations above, where the quadratic equation satisfied by an OGF is amenable to an explicit solution-the resulting OGF could then be expanded by means of Newton’s binomial theorem. Similarly, we derive later in this Chapter an explicit form for the number of integer compositions by means of the symbolic method and OGFs the answer turns out to be simply $2^{n-1}$ ) and derive many explicit specializations. In this book, we assume as known the elementary techniques from basic calculus by which the Taylor expansion of an explicitly given function can be obtained. Good references on such elementary aspects are Wilf’s Generatingfunctionology [406], Graham, Knuth, and Patashnik’s Concrete Mathematics [196], and our book [353].

## 数学代写|组合学代写Combinatorics代考|Implicit enumeration formulæ

Implicit enumeration formula. In a number of cases, the generating functions obtained by the symbolic method are still in a sense explicit, but their form is such that their coefficients are not clearly reducible to a closed form. It is then still possible to obtain initial values of the corresponding counting sequence by means of a symbolic manipulation system. Also, from generating functions, it is possible to derive systematically recurrences $^2$ that lead to a procedure for computing an arbitrary number of terms of the counting sequence in a reasonably efficient manner. A typical example of this situation is the OGF of integer partitions,
$$P(z)=\prod_{m=1}^{\infty} \frac{1}{1-z^m},$$
for which recurrences obtained from the $\mathrm{OGF}$ and associated to fast algorithms are given in Note 12 (p. 39) and Note 17 (p. 46).

## 数学代写|组合学代写Combinatorics代考|Implicit enumeration formulæ

$$P(z)=\prod_{m=1}^{\infty} \frac{1}{1-z^m}$$

## 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。

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## 数学代写|组合学代写Combinatorics代考|Admissible constructions and specifications

The main goal of this section is to introduce formally the basic constructions that constitute the core of a specification language for combinatorial structures. This core is based on disjoint unions, also known as combinatorial sums, and on Cartesian products that we have just discussed. We shall augment it by the constructions of sequence, cycle, multiset, and powerset. A class is constructible or specifiable if it can be defined from primal elements by means of these constructions. The generating function of any such class satisfies functional equations that can be transcribed systematically from a specification; see Theorems I.1 and I.2, as well as Figure 14 at the end of this chapter for a summary.

I. 2.1. Basic constructions. First, we assume given a class $\mathcal{E}$ called the neutral class that consists of a single object of size 0 ; any such an object of size 0 is called a neutral object. and is usually denoted by symbols like $\epsilon$ or 1 . The reason for this terminology becomes clear if one considers the combinatorial isomorphism
$$\mathcal{A} \cong \mathcal{E} \times \mathcal{A} \cong \mathcal{A} \times \mathcal{E}$$

## 数学代写|组合学代写Combinatorics代考|Combinatorial sum (disjoint union)

Combinatorial sum (disjoint union). First consider combinatorial sum also known as disjoint union. The intent is to capture the union of disjoint sets, but without the constraint of any extraneous condition of disjointness. We formalize the (combinatorial) sum of two classes $\mathcal{B}$ and $\mathcal{C}$ as the union (in the standard set-theoretic sense) of two disjoint copies, say $\mathcal{B}^{\square}$ and $\mathcal{C}^{\diamond}$, of $\mathcal{B}$ and $\mathcal{C}$. A picturesque way to view the construction is as follows: first choose two distinct colours and repaint the elements of $\mathcal{B}$ with the $\square$-colour and the elements of $\mathcal{C}$ with the $\diamond$-colour. This is made precise by introducing two distinct “markers” $\square$ and $\diamond$, each a neutral object (i.e., of size zero); the disjoint union $\mathcal{B}+\mathcal{C}$ of $\mathcal{B}, \mathcal{C}$ is then defined as the standard set-theoretic union,
$$\mathcal{B}+\mathcal{C}:=({\square} \times \mathcal{B}) \cup({\diamond} \times \mathcal{C})$$
The size of an object in a disjoint union $\mathcal{A}=\mathcal{B}+\mathcal{C}$ is by definition inherited from its size in its class of origin, like in Equation (13). One good reason behind the definition adopted here is that the combinatorial sum of two classes is always welldefined. Furthermore, disjoint union is equivalent to a standard union whenever it is applied to disjoint sets.

## 数学代写|组合学代写Combinatorics代考|Admissible constructions and specifications

$$\mathcal{A} \cong \mathcal{E} \times \mathcal{A} \cong \mathcal{A} \times \mathcal{E}$$

## 数学代写|组合学代写Combinatorics代考|Combinatorial sum(disjoint union)

$$\mathcal{B}+\mathcal{C}:=(\square \times \mathcal{B}) \cup(\diamond \times \mathcal{C})$$

## 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。

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## avatest™帮您通过考试

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## 数学代写|组合学代写Combinatorics代考|Hubert’s single-link algorithm

Given a set of $n$ objects $X=\left{x_1, x_2, \ldots, x_n\right}$, the dissimilarity table, and a threshold value $\lambda$.

Set $m=0$ and form the disjoint clustering of zero level,
$$\mathbf{C}0=\left{\mathcal{C}{0,1}, \mathcal{C}{0,2}, \ldots, \mathcal{C}{0, n}\right}$$
consisting of $n$ 1-element clusters $\mathcal{C}{0, k}=\left{x_k\right}, k=1, \ldots, n$. Define the function $S_0$ and the dissimilarities between the clusters of level zero by $$S_0(a, b)=\operatorname{diss}\left(\mathcal{C}{0, a}, \mathcal{C}{0, b}\right)=d\left(x_a, x_b\right) .$$ Find the minimum value $S_0^{\min }$ of the function $S_0(a, b)$ over all the pairs $(a, b)$ $$S_0^{\min }=\min {a, b} S_0(a, b)=S_0(p, q),$$
attained at the pair $(p, q)$. This pair indicates the clusters of zeroth level, $\mathcal{C}{0, p}$ and $\mathcal{C}{0, q}$, to be merged in a cluster of the first level,
$$\mathcal{C}{1,1}=\mathcal{C}{0, p} \cup \mathcal{C}{0, p}$$ All the other zeroth-level clusters remain the same, we only have to renumber them, $$\mathcal{C}{1, r}=\mathcal{C}_{0, s}, \quad r \geq 2, s \neq p, s \neq q .$$

## 数学代写|组合学代写Combinatorics代考|Hubert’s complete-link algorithm

In this section we consider a different approach to amalgamated clustering, called complete-link clustering. An essential distinction between the single-link and complete-link algorithms is the rule of merging two existing clusters into one of a higher level. Instead of connected subgraphs of the threshold graph $G(\infty)$ used in the single linkage, now we consider the maximum complete subgraphs of $G(\infty)$. Examples show that the single linkage and the complete linkage may result in different clusterings.
Coffee-time browsing

• www.sigkdd.org/explorations/issue4-1/estivill.pdf
We are concerned with another Hubert’s clustering algorithm called complete-link clustering [31]. We use the same notations as in the previous sections, but consider only dissimilarity matrices without ties. ${ }^2$ We again start with an informal description of the algorithm and then write down its pseudo-code.

Like the single linkage, the complete linkage uses the same sequence of the threshold graphs. To avoid any ambiguity, we denote complete-link clusterings by $\mathbf{C}m^{\text {comp }}$. Given a clustering $$\mathbf{C}_m^{\mathrm{comp}}=\left{\mathcal{C}{m, 1}, \mathcal{C}{m, 2}, \ldots, \mathcal{C}{m, n_m}\right}$$
of the $m$ th level, $m=0,1,2, \ldots$, we consider all pairwise unions
$$\mathcal{C}{m, a} \cup \mathcal{C}{m, b}, \quad a, b=1,2, \ldots, n_m, a \neq b$$

## 数学代写|组合学代写Combinatorics代考|Hubert’s single-link algorithm

\left 缺少或无法识别的分隔符

$$S_0(a, b)=\operatorname{diss}(\mathcal{C} 0, a, \mathcal{C} 0, b)=d\left(x_a, x_b\right) .$$

$$S_0^{\min }=\min a, b S_0(a, b)=S_0(p, q),$$

$$\mathcal{C} 1,1=\mathcal{C} 0, p \cup \mathcal{C} 0, p$$

$$\mathcal{C} 1, r=\mathcal{C}_{0, s}, \quad r \geq 2, s \neq p, s \neq q .$$

## 数学代写|组合学代写Combinatorics代考|Hubert’s complete-link algorithm

• www.sigkdd.org/explorations/issue4-1/estivill.pdf
我们关注另一种称为完全链接聚类 [31] 的 Hubert 聚类算法。我们使用与前面部分相同的符号，但只考 虑没有关系的相异矩阵。 ${ }^2$ 我们再次从算法的非正式描述开始，然后写下它的伪代码。
与单链接一样，完整链接使用相同的阈值图序列。为了避免歧义，我们将完整链接聚类表示为 $\mathbf{C} m^{\text {comp }}$. 给定 一个聚类
\left 缺少或无法识别的分隔符
的 $m$ 第级， $m=0,1,2, \ldots$ ，我们考虑所有成对联合
$$\mathcal{C} m, a \cup \mathcal{C} m, b, \quad a, b=1,2, \ldots, n_m, a \neq b$$

## 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。

Posted on Categories:Combinatorics, 数学代写, 组合学, 组合数学

## avatest™帮您通过考试

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## 数学代写|组合数学代写Combinatorial Mathematics代考|Existence and non-existence of tight designs

Let $V={1,2, \ldots, v}$, and let $X=\left(\begin{array}{l}V \ d\end{array}\right)$ be the set of $d$-element subsets of $V$. We assume $1 \leq t \leq d \leq \frac{v}{2}$. Let $\left(X,\left{R_i\right}_{0 \leq i \leq d}\right)$ be the Johnson scheme $J(v, d)$. Recall that a 2 -design $Y \subset X$ is tight if $\left.|Y|=m_0+m_1+\cdots+m_e=\sum_{i=0}^e\left(\begin{array}{l}v \ i\end{array}\right)-\left(\begin{array}{c}v \ i-1\end{array}\right)\right)=\left(\begin{array}{c}v \ e\end{array}\right)$ holds. In general, a 2e-design $Y$ satisfies the Fisher type inequality $|Y| \geq\left(\begin{array}{l}v \ v\end{array}\right)$. This inequality was first obtained by Petrenjuk [395] for the case $e=2$. Afterwards, Ray-Chaudhuri and Wilson announced that it holds for any $e$ [514]. For the proof, see Ray-Chaudhuri and Wilson [400]. Delsarte (1974) also gave a proof, and as was written in the book of Hiroshi Nagao [366], Noda and Bannai obtained the same result independently in 1972 . If there exists a tight $2 e$-design, by using Theorem 3.16, we can show the following theorem.
Theorem 3.27. If there exists a tight $2 e$-design in the Johnson scheme $J(v, d)$, then all $e$ zeros of the polynomial
$$\Psi_e(x)=\sum_{i=0}^e(-1)^{e-i} \frac{\left(\begin{array}{c} v-e \ i \end{array}\right)\left(\begin{array}{c} k-i \ e-i \end{array}\right)\left(\begin{array}{c} k-1-i \ e-i \end{array}\right)}{\left(\begin{array}{c} e \ i \end{array}\right)}\left(\begin{array}{c} x \ i \end{array}\right)$$
are positive integers. This polynomial $\Psi_e$ is called the Wilson polynomial or the RayChaudhuri-Wilson polynomial.

When $e=1, Y$ is a tight 2-design if and only if $b=v(b=|Y|)$ if and only if $Y$ is a symmetric 2-design (Chapter 1, Section 1.3, Definition 1.37). There exist quite a few symmetric 2-designs and their classification seems to be almost impossible.

When $e=2$, the non-trivial tight 4-designs are the Witt design 4- $(23,7,1)$ and its complementary design 4- $(23,16,52)$ only. (In the latter case, $d \leq \frac{v}{2}$ does not hold.) The classification was started by Noboru Ito $[245,246]$ and was almost completed by Enomoto, Ito, and Noda [179]. To be precise, the classification was completely solved by Bremner [111] and Stroeker [440] by determining a rational integral solution of the Diophantine equation $3 x^4-4 y^4-2 x^2+12 y^2-9=0$, which is related to an elliptic function. In the next subsection, we present the detailed proof by Noda.

## 数学代写|组合数学代写Combinatorial Mathematics代考|Classiffcation of tight 4-designs in Johnson schemes

Let $(V, \mathcal{B})$ be a tight 4-design in the Johnson scheme $J(v, k)$. Suppose $(V, \mathcal{B})$ is nontrivial. Assume $k \leq \frac{v}{2}$. Note that if $k>\frac{v}{2}$, the complementary design is also a tight 4-design. In this section, we prove the following theorem.

Theorem $3.32$ (Enomoto-Ito-Noda [179]). A non-trivial tight 4-design in the Johnson scheme is the 4- $(23,7,1)$ design or its complementary design 4-(23, 16, 52) only.

Remark 3.33. Noboru Ito $[245,246]$ started the proof of this theorem, and Enomoto, Ito, and Noda (1979) [179] almost completed the proof by correcting errors. In a part of the proof, a number theoretic result on the solution of a Diophantine equation was used (Bremner [111], Stroeker [440]). The proof given here is based on an unpublished note by Ryuzaburo Noda, which was written soon after [179]. We are grateful to Professor Noda, who permits us to use the contents of his note. It is similar to [179] that the problem is transformed into the Diophantine equation. However, compared to the proof combining three papers $[245,246,179]$, this proof is clearer and easier to read.
The proof of Theorem $3.32$ consists of steps $(\mathrm{A})-(\mathrm{K})$.
(A) Let $i, j$ be the cardinalities of intersections of two distinct blocks. Let $i<j$. Then $i, j$ are the roots of the following quadratic equation:
$$X^2-\left(\frac{2(k-1)(k-2)}{v-3}+1\right) X+\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}=0 .$$
(B) We have
$$(v-2)(v-3) \mid 2 k(k-1)(k-2)$$
Proof. Since $b=\lambda_0=\left(\begin{array}{c}v \ 2\end{array}\right), \lambda_4=\frac{k(k-1)(k-2)(k-3)}{2(v-2)(v-3)}$ is an integer. Therefore, $2 \lambda_4=$ $\frac{k(k-1)(k-2)(k-3)}{(v-2)(v-3)}$ is an integer. On the other hand, by (A), $\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}$ is an integer, and hence $\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}-\frac{k(k-1)(k-2)(k-3)}{(v-2)(v-3)}=\frac{2 k(k-1)(k-2)}{(v-2)(v-3)}$ is also an integer.

## 数学代写|组合数学代写组合数学代考|紧密设计的存在与不存在

$$\Psi_e(x)=\sum_{i=0}^e(-1)^{e-i} \frac{\left(\begin{array}{c} v-e \ i \end{array}\right)\left(\begin{array}{c} k-i \ e-i \end{array}\right)\left(\begin{array}{c} k-1-i \ e-i \end{array}\right)}{\left(\begin{array}{c} e \ i \end{array}\right)}\left(\begin{array}{c} x \ i \end{array}\right)$$

## 数学代写|组合数学代写组合数学代考| Johnson方案中紧密的4-设计的分类

(A)设$i, j$为两个不同块的交集的基数。让$i<j$。那么$i, j$是下面的二次方程的根:
$$X^2-\left(\frac{2(k-1)(k-2)}{v-3}+1\right) X+\frac{k(k-1)^2(k-2)}{(v-2)(v-3)}=0 .$$
(B)我们有
$$(v-2)(v-3) \mid 2 k(k-1)(k-2)$$

## 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。