## 数学代写|组合学代写Combinatorics代考|CS-E4555

2023年3月22日

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## 数学代写|组合学代写Combinatorics代考|d-regular partitions

After seeing how the Stirling numbers are related to the enumeration problem of functions, we study another question ${ }^{20}$.

Once upon a time, a mad king of a small kingdom of five cities was facing a problem. The citizens of the capital did not want to pay his newly introduced air tax. The king knew that there were some rebels convincing the citizens not to pay the tax. He also knew that a group of six rebels were causing the problem who were living in adjacent houses. The king decided to separate the instigators: he announced that all of these rebels must be moved to five other cities. To make it even harder for them to form a group again, the king added that none of two adjacent neighbors can move to the same city. How many possibilities did the gendarmerie have to execute the king’s order?

Since there are six rebellious instigators, but only five cities, two of the rebels will surely go to the same city. The first rebel can live together with the third, fourth, fifth, or sixth rebel, and the others are separated. These give four possibilities. Or, the second rebel can live together with one of the fourth, fifth, or sixth rebels, the third rebel together with the fifth or sixth (here we do not write the first-third pairing, because this configuration was counted before). Finally, the fourth rebel can live together with the sixth (and also with the already counted first and second) rebel. These give $4+3+2+1=10$ cases in total. The question has been answered.

How can we generalize this problem? The adjacent neighbors could not be put together. Therefore, a possible direction of generalization is that we restrict the citizens to live together only with even more distant neighbors. We introduce the following definition.

## 数学代写|组合学代写Combinatorics代考|Zigzag permutations

Applying some restriction on the combinatorial structures at hand often gives interesting constructions. We saw one example already, the $d$-regular partitions. In that instance, the restriction (that the elements must have some distance greater than one in between, resulted in no new number sequence because the $d$-regular partitions can be counted by the usual Stirling numbers of the second kind (see (1.14)).

Here we introduce an important subclass of permutations when the elements must be in a “zigzag” form. This definition will result in a new and interesting counting sequence. For example,
$$\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \ 3 & 1 & 4 & 2 & 5 \end{array}\right)$$
is a “zigzag” permutation (more precisely, a down-up zigzag), because the elements in the bottom line can be ordered in a zigzag form:
$$3>1<4>2<5 .$$ Here goes the general definition. Definition 1.8.1. A permutation $$\left(\begin{array}{ccccc} 1 & 2 & 3 & \cdots & n \\ i_1 & i_2 & i_3 & \cdots & i_n \end{array}\right)$$ is called up-down zigzag if $$i_1>i_2\cdots .$$
Similarly, this permutation is down-up zigzag if, instead, their elements satisfy the inequalities
$$i_1i_3<\cdots .$$

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Zigzag permutations

$$\left(\begin{array}{llll} 1 & 2 & 3 & 4 & 5 \ 3 & 1 & 4 & 2 & 5 \end{array}\right)$$

$$3>1<4>2<5 。$$ 这是一般定义。定义 1.8.1。排列 $$\left(\begin{array}{ccccc} 1 & 2 & 3 & \cdots & n \\ i_1 & i_2 & i_3 & \cdots & i_n \end{array}\right)$$ 被调用- 如果 $$i_1>i_2\cdots 向下之字形。$$类似地，如果相反，它们的元素满足不等式 i_1i_3<\cdots ，则此排列是向下的锯齿形

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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。