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• 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

## 英国补考|组合学代写Combinatorics代考|Classifying All Partitions

First, we know the generating function of partitions with exactly $m$ parts. This family is obtained from partitions with at most $m$ parts by adding a column of $m$ cells. It follows from (2.44) that the generating function of partitions with exactly $m$ parts is
$$\frac{q^{m}}{(1-q)\left(1-q^{2}\right) \cdots\left(1-q^{m}\right)}$$
This then must be the coefficient of $x^{m}$ in the expansion of (2.46). In other words
$$\prod_{m \geq 1} \frac{1}{1-x q^{m}}=1+\sum_{m \geq 1} \frac{x^{m} q^{m}}{(1-q)\left(1-q^{2}\right) \cdots\left(1-q^{m}\right)}$$
Secondly, we can generate all partitions according to the number of parts by classifying them according to the size of their Durfee square. We do this as follows.
From the decomposition in Figure 2.33, we can recover the number of parts of $\lambda$ by adding to $m$ the number of parts of $\mu$. But the generating function for the $\mu$ ‘s which keeps track of the number of parts as the exponent of $x$ is
$$\frac{1}{(1-x q)\left(1-x q^{2}\right) \cdots\left(1-x q^{m}\right)}$$
Clearly each part used to build up $\mu$ contributes 1 to the exponent of $x$ in the expansion of this. We have no restriction on the $\tau$ ‘s attached to the right of the Durfee square other than they should not have more than $m$ parts. The generating function of the $\tau$ ‘s is of course (2.44).

## 英国补考|组合学代写Combinatorics代考|Self-Conjugate Partitions

Another application of the Durfee square idea is the enumeration of self-conjugate partitions. These are partitions which are equal to their own conjugate, in other words partitions whose diagram is symmetric with respect to the diagonal line $y=x$

We will construct the generating function of self-conjugate partitions two different ways.

Again consider the decomposition of a partition into the three pieces consisting of an $m \times m$ square and the two partitions $\tau$ and $\mu$ as we did in Figure 2.33.

However, now the $\tau$ and $\mu$ in decomposition of Figure $2.33$ must be so that $\tau=\mu^{\prime}$ since we need to fall back on the same partition after reflecting about the diagonal line (Fig. 2.35).

This means that instead of using Figure $2.34$ to generate the relevant partitions to add to the Durfee square, now we need to pick pairs of identical parts as generated symbolically by the expression in Figure 2.36. Here one of the double rows picked is used as a row of $\mu$, while the other is used as a column of $\tau$. We see that the generating function for self-conjugate partitions which have an $m \times m$ Durfee square is
$$\frac{q^{m^{2}}}{\left(1-q^{2}\right)\left(1-q^{4}\right) \cdots\left(1-q^{2 m}\right)}$$
Summing over all $m$, the generating function of self-conjugate partitions is given by
$$1+\sum_{m \geq 1} \frac{q^{m^{2}}}{\left(1-q^{2}\right)\left(1-q^{4}\right) \cdots\left(1-q^{2 m}\right)}$$

# 组合学代考

## 英国补考|组合学代写Combinatorics代考|Classifying All Partitions

$$\frac{q^{m}}{(1-q)\left(1-q^{2}\right) \cdots\left(1-q^{m}\right)}$$

$$\prod_{m \geq 1} \frac{1}{1-x q^{m}}=1+\sum_{m \geq 1} \frac{x^{m} q^{m}}{(1-q)\left(1-q^{2}\right) \cdots\left(1-q^{m}\right)}$$

$$\frac{1}{(1-x q)\left(1-x q^{2}\right) \cdots\left(1-x q^{m}\right)}$$

## 英国补考|组合学代写Combinatorics代考|Self-Conjugate Partitions

Durfee 平方思想的另一个应用是自共轭分区的枚举。这些是等于它们自己的共轭的分区，换句话说，其图关于对角线对称的分区 $y=x$

$$\frac{q^{m^{2}}}{\left(1-q^{2}\right)\left(1-q^{4}\right) \cdots\left(1-q^{2 m}\right)}$$

$$1+\sum_{m \geq 1} \frac{q^{m^{2}}}{\left(1-q^{2}\right)\left(1-q^{4}\right) \cdots\left(1-q^{2 m}\right)}$$

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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。

assignmentutor™您的专属作业导师
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