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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

## 数学代写|图论作业代写Graph Theory代考|Stirling Permutations

Stirling permutations were introduced by Gessel and Stanley in [7]. Denoted by $\mathcal{Q}{n}$, they are defined as those permutations $\pi{1} \pi_{2} \ldots \pi_{2 n}$ of the multiset ${1,1,2,2, \ldots, n, n}$ satisfying that, if $i\pi_{i}$. This condition can be described as avoiding the pattern 212. In general, given two sequences of positive integers $\pi$ and $\sigma$, we say that $\pi$ avoids $\sigma$ if there is no subsequence of $\pi$ whose entries are in the same relative order as those of $\sigma$. There is an extensive literature on Stirling permutations and their generalizations $[3,4,8,9,12]$.

With the notation $[r]={1,2, \ldots, r}$, let $i \in[r]$ be a descent of $\pi=\pi_{1} \pi_{2} \ldots \pi_{r}$ if $\pi_{i}>\pi_{i+1}$ or $i=r$, and let $\operatorname{des}(\pi)$ denote the number of descents of $\pi$. This definition agrees with $[3,7,9]$, while other papers [2] do not consider $i=r$ to be a descent. We also consider ascents, which are indices $i \in{0, \ldots, r-1}$ such that $\pi_{i}<\pi_{i+1}$ or $i=0$, and plateaus, which are indices $i \in[r-1]$ such that $\pi_{i}=\pi_{i+1}$. Let $\operatorname{asc}(\pi)$ and plat $(\pi)$ denote the number of ascents and the number of plateaus of $\pi$, respectively.
Denoting by $\mathcal{S}{n}$ the set of permutations of $[n]$, the polynomials $$A{n}(t)=\sum_{\pi \in \mathcal{S}{n}} t^{\mathrm{des}(\pi)}$$ are called Eulerian polynomials. It is well known (see e.g. [13, Prop. 1.4.4]) that $$\sum{m \geq 0} m^{n} t^{m}=\frac{A_{n}(t)}{(1-t)^{n+1}}$$
Let $S(n, m)$ be the Stirling numbers of the second kind, which count partitions of an $n$-element set into $m$ blocks. Gessel and Stanley [7] showed that, when replacing the coefficients in the left-hand side of Eq. (2) by these numbers, then the role of the Eulerian polynomials is played by the Stirling polynomials $Q_{n}(t)=\sum_{\pi \in \mathcal{Q}_{n}} t^{\operatorname{des}(\pi)}$

## 数学代写|图论作业代写Graph Theory代考|Quasi-Stirling Permutations

In [2], Archer et al. introduce the set $\overline{\mathcal{Q}}{n}$ of quasi-Stirling permutations. These are permutations $\pi{1} \pi_{2} \ldots \pi_{2 n}$ of the multiset ${1,1,2,2, \ldots, n, n}$ avoiding 1212 and 2121 , which means that there do not exist $i<j<k<\ell$ such that $\pi_{i}=\pi_{k}$ and $\pi_{j}=\pi_{\ell}$. Thinking of $\pi$ as a labeled matching of $[2 n]$, by placing an arc between with label $k$ between $i$ with $j$ if $\pi_{i}=\pi_{j}=k$, the avoidance requirement is equivalent to the matching being noncrossing ${ }^{1}$. By definition, $\mathcal{Q}{n} \subseteq \overline{\mathcal{Q}}{n}$.
Archer et al. [2] note that $\left|\overline{\mathcal{Q}}{n}\right|=n ! C{n}=\frac{(2 n) !}{(n+1) !}$, where $C_{n}=\frac{1}{n+1}\left(\begin{array}{c}2 n \ n\end{array}\right)$ is the $n$th Catalan number. They also compute the number of permutations in $\overline{\mathcal{Q}}{n}$ avoiding some sets of patterns of length 3, and they enumerate quasistifling permutateng by the numbef of plateaus, They pgge the spen prgblem of enumerating quasi-Stirling permutations by the number of descents, and they conjecture that the number of $\pi \in \overline{\mathcal{Q}}{n}$ with $\operatorname{des}(\pi)=n$ is equal to $(n+1)^{n-1}$.

## 数学代写|图论作业代写Graph Theory代考|Stirling Permutations

$$A n(t)=\sum_{\pi \in \mathcal{S}{n}} t^{\mathrm{des}(\pi)}$$ 称为欧拉多项式。众所周知 (参见例如 [13, Prop. 1.4.4]) $$\sum m \geq 0 m^{n} t^{m}=\frac{A{n}(t)}{(1-t)^{n+1}}$$

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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
assignmentutor™您的专属作业导师