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Restricted constructions. An immediate formula for OGFs is that of the diagonal $\Delta$ of a cartesian product $\mathcal{B} \times \mathcal{B}$ defined as
$$\mathcal{A} \equiv \Delta(\mathcal{B} \times \mathcal{B}):={(\beta, \beta) \mid \beta \in \mathcal{B}}$$
Then, clearly $A_{2 n}=B_n$ so that
$$A(z)=B\left(z^2\right)$$
The diagonal construction permits us to access the class of all unordered pairs of (distinct) elements of $\mathcal{B}$, which is $\mathcal{A}=\operatorname{PSET}_2(\mathcal{B})$. A direct argument then runs as follows: the unordered pair ${\alpha, \beta}$ is associated to the two ordered pairs $(\alpha, \beta)$ and $(\beta, \alpha)$ except when $\alpha=\beta$, where an element of the diagonal is obtained. In other words, one has the combinatorial isomorphism,
$$\operatorname{PSET}_2(\mathcal{B})+\operatorname{PSET}_2(\mathcal{B})+\Delta(B \times B) \cong B \times B$$
meaning that
$$2 A(z)+B\left(z^2\right)=B(z)^2$$
The resulting translation into OGFs is thus
$$\mathcal{A}=\operatorname{PSET}_2(\mathcal{B}) \quad \Longrightarrow \quad A(z)=\frac{1}{2} B(z)^2-\frac{1}{2} B\left(z^2\right)$$
Similarly, for multisets, we find
$$\mathcal{A}=\operatorname{MSET}_2(\mathcal{B}) \quad \Longrightarrow \quad A(z)=\frac{1}{2} B(z)^2+\frac{1}{2} B\left(z^2\right)$$
while for cycles one has $\mathrm{CYC}_2 \cong \mathrm{MSET}_2$, and
$$\mathcal{A}=\mathrm{CYC}_2(\mathcal{B}) \quad \Longrightarrow \quad A(z)=\frac{1}{2} B(z)^2+\frac{1}{2} B\left(z^2\right)$$

## 数学代写|组合学代写Combinatorics代考|Pointing and substitution

Pointing and substitution. Two more constructions, namely pointing and substitution, translate agreeably into generating functions. Combinatorial structures are viewed here as formed of “atoms” (words are composed of letters, graphs of nodes, etc) which determine their sizes. In this context, pointing means “pointing at a distinguished atom”; substitution, written $\mathcal{B} \circ \mathcal{C}$ or $\mathcal{B}[\mathcal{C}]$, means “substitute elements of $\mathcal{C}$ for atoms of $\mathcal{B}$ “.

DEFINITION I.14. Let $\left{\epsilon_1, \epsilon_2, \ldots\right}$ be a fixed collection of distinct neutral objects of size 0. The pointing of a class $\mathcal{B}$, noted $\mathcal{A}=\Theta \mathcal{B}$, is formally defined by
$$\Theta \mathcal{B}:=\sum_{n \geq 0} \mathcal{B}n \times\left{\epsilon_1, \ldots, \epsilon_n\right}$$ The substitution of into $\mathcal{B}$ (also known as composition of $\mathcal{B}$ and $\mathcal{C}$ ), noted $\mathcal{B} \circ \mathcal{C}$ or $\mathcal{B}[\mathcal{C}]$, is formally defined as $$\mathcal{B} \circ \mathcal{C} \equiv \mathcal{B}[\mathcal{C}]:=\sum{k \geq 0} \mathcal{B}_k \times \mathrm{SEQ}_k(\mathcal{C})$$
If $B_n$ is the number of $\mathcal{B}$ structures of size $n$, then $n B_n$ can be interpreted as counting pointed structures where one of the $n$ atoms composing a $\mathcal{B}$-structure has been distinguished (here by a special “pointer” of size 0 attached to it). Elements of $\mathcal{B} \circ \mathcal{C}$ may also be viewed as obtained by selecting in all possible ways an element $\beta \in \mathcal{B}$ and replacing each of its atoms by an arbitrary element of $\mathcal{C}$.

## 组合学代写

$$\mathcal{A} \equiv \Delta(\mathcal{B} \times \mathcal{B}):=(\beta, \beta) \mid \beta \in \mathcal{B}$$

$$A(z)=B\left(z^2\right)$$

$$\operatorname{PSET}_2(\mathcal{B})+\operatorname{PSET}_2(\mathcal{B})+\Delta(B \times B) \cong B \times B$$

$$2 A(z)+B\left(z^2\right)=B(z)^2$$

$$\mathcal{A}=\operatorname{PSET}_2(\mathcal{B}) \quad \Longrightarrow \quad A(z)=\frac{1}{2} B(z)^2-\frac{1}{2} B\left(z^2\right)$$

$$\mathcal{A}=\operatorname{MSET}_2(\mathcal{B}) \quad \Longrightarrow \quad A(z)=\frac{1}{2} B(z)^2+\frac{1}{2} B\left(z^2\right)$$

$$\mathcal{A}=\mathrm{CYC}_2(\mathcal{B}) \quad \Longrightarrow \quad A(z)=\frac{1}{2} B(z)^2+\frac{1}{2} B\left(z^2\right)$$

## 数学代写|组合学代写Combinatorics代考|Pointing and substitution

|Theta $\backslash$ mathcal ${B}:=\backslash$ sum_{n $\backslash$ geq 0$} \backslash$ mathcal ${B} n \backslash$ times $\backslash$ left ${$ lepsilon_I, \Idots, \epsilon_n $\backslash$ right $}$

$$\mathcal{B} \circ \mathcal{C} \equiv \mathcal{B}[\mathcal{C}]:=\sum k \geq 0 \mathcal{B}_k \times \mathrm{SEQ}_k(\mathcal{C})$$

## 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。