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数学代写|组合学代写Combinatorics代考|Paths in a Rectangle

A network of roads constructed in two mutually orthogonal directions is an inexhaustible source of interesting and instructive combinatorial problems, which are the topic of the current chapter. Probably, the most prominent among them is presented below in the first paragraph.

  1. Consider an $m \times n$ rectangle split into $m$ vertical and $n$ horizontal stripes by straight lines parallel to its sides. In Fig. 4.1, there is an illustration of the above for $m=7$ and $n=4$

Intersecting pairwise, these stripes create $m n$ (in Fig. 4.1 there are 35 of them) squares with sides of length 1 . Imagine that all lines ( $m+1$ vertical and $n+1$ horizontal) are paths, following which one can get, say, from the “southwestern” point $A$ to the “northeastern” point $B$. Departing from the point $A$ with the intention to get to the point $B$ following the shortest possible route, at each intersection, one has to choose where to move next. Is there a rule, following which one can get from $A$ to $B$ using the shortest eligible path and not get lost in the way? Obviously, one such path can be pointed out (e.g., $A C B$ ). However, the problem is not about this. What we actually need is to define features of the shortest paths that distinguish them from all other possible routes along the drawn lines. First, we need to find the length of the shortest path from $A$ to $B$. This is rather simple task. We are allowed to move along the horizontal and vertical lines only, that is, we are limited to two orthogonal directions: “south-north” and “west-east”. The point $B$ is located in $m$ units of length to the east and $n$ units of length to the north from the point $A$. Thus, in order to get to the destination, we have to cover the distance of $m$, moving from the west to the east, and the distance of $n$, moving from the south to the north. This means that the shortest paths from $A$ to $B$ are of length $m+n$. For example, $A C B$ and $A D B$ are two of such paths. But there are other paths. A path from $A$ to $B$ is the shortest if it consists of $m$ horizontal and $n$ vertical elementary (of length 1) intervals (this is how we will call the part of path between two adjacent points of intersection). It does not matter in which order one passes along these intervals. That is why there are many shortest paths from $A$ to $B$. In Fig. $4.1$ two of them are outlined. The first and the most important question about paths in a rectangle concerns to the number of the shortest paths from the point $A$ to the point $B$.

数学代写|组合学代写Combinatorics代考|Graphs of functions located on intersecting lines

Draw six straight lines on the coordinate plane (see Fig. 4.12)
$$
y=x+1, y=x, y=x-1, y=-x+1, y=-x, y=-x-1 .
$$
Points of intersection of three first of them with three others form a square $P Q T S$, which is split by the middle lines into 4 smaller squares. Consider these six lines with no regard to the coordinate axes. In the combined graph of the above lines. we will further call the line L.
We have to answer the following question:
How many different functions possess three following properties:
a) a function is defined on $R$ (the set of all real numbers);
b) it is continuous (there are no discontinuities on its graph);
c) its graph is part of the line $L$ ?
Recall that an arbitrary line on the coordinate plane can not be a graph of a function unless any vertical straight line (the line $x=c$ ) crosses have at most one point of intersection with it. If a function is defined on the set of all real numbers (on $R$ ), then each vertical line intersects with its graph in one point. In view of properties b) and c) how does this fact affect the structure of a graph in our case? Which part of $L$ it can be, and which it can not? The graph of every our function is a polygonal chain (or a straight line) which “comes” from the north or northwest, travels inside the square $P Q T S$, and vanishes in the north- or south-eastern direction. Imagine a point, which appears on the left ray of this polygonal chain and moves to the right. Then the vector of the velocity of this point is always directed to the northeast or southeast, changing its direction in the vertices of the polygonal chain.
All six graphs of initial functions are among the wanted ones. All other graphs are polygonal chains composed of two (left and right) rays and, possibly, several line segments from the square $P Q T S$. Here are some of these polygonal chains (see Fig. 4.13): $K M Q N F, K M Q T E, K O N J, H N T E, H N J, V P S T J, V M O F, V M O N T E, V Q T E$.

So how many such polygonal chains are there in total? Arranging the process of search, finding enough patience and concentration, we could eventually derive the number of graphs in this case, where the line $L$ is constructed with six lines. However, such a result would not have much value, because solving this type of special case, it is desirable to bear in mind their natural generalizations.

数学代写|组合学代写Combinatorics代考|MATH393

组合学代写

数学代写|组合学代写Combinatorics代考|Paths in a Rectangle

在两个相互正交的方向上构建的道路网络是有趣和有启发性的组合问题的取之不尽的来源,这也是本章的主题。其中最突出的可能在第一段中介绍。

  1. 考虑一个米×n矩形分割成米垂直和n由平行于其两侧的直线形成的水平条纹。在图 4.1 中,有上面的说明米=7和n=4

成对相交,这些条纹创建米n(在图 4.1 中有 35 个)边长为 1 的正方形。想象一下所有行(米+1垂直和n+1水平) 是路径,沿着这些路径可以从“西南”点到达一个到“东北”点乙. 从点出发一个目的是要直截了当乙沿着最短的路线,在每个交叉路口,人们必须选择下一步要移动到哪里。有没有一个规则,可以从中得到一个至乙使用最短的合格路径并且不会迷路?显然,可以指出一条这样的路径(例如,一个C乙)。然而,问题不在于这个。我们真正需要的是定义最短路径的特征,将它们与沿着绘制的线的所有其他可能路径区分开来。首先,我们需要找到最短路径的长度一个至乙. 这是一个相当简单的任务。我们只允许沿着水平线和垂直线移动,也就是说,我们被限制在两个正交方向:“南-北”和“西-东”。重点乙位于米向东的长度单位和n从该点向北的长度单位一个. 因此,为了到达目的地,我们必须覆盖米,从西向东移动,距离n,从南向北移动。这意味着最短路径从一个至乙有长度米+n. 例如,一个C乙和一个D乙是两条这样的路径。但还有其他途径。一条从一个至乙是最短的,如果它由米水平和n垂直基本(长度为 1)间隔(这就是我们如何称呼两个相邻交点之间的路径部分)。通过这些间隔的顺序无关紧要。这就是为什么有许多最短路径从一个至乙. 在图。4.1其中两个被概述。关于矩形路径的第一个也是最重要的问题涉及到从该点开始的最短路径的数量一个切中要害乙.

数学代写|组合学代写Combinatorics代考|Graphs of functions located on intersecting lines

在坐标平面上画六条直线(见图 4.12)

是=X+1,是=X,是=X−1,是=−X+1,是=−X,是=−X−1.
其中三个与其他三个的交点形成一个正方形磷问吨小号, 它被中间线分成 4 个较小的正方形。考虑这六行而不考虑坐标轴。在上述线条的组合图中。我们将进一步称为 L 行。
我们必须回答以下问题:
有多少不同的函数具有以下三个属性:
a) 函数定义在R(所有实数的集合);
b) 它是连续的(图上没有间断);
c) 它的图形是直线的一部分大号?
回想一下,坐标平面上的任意一条线都不能是函数的图,除非有任何垂直的直线(线X=C) 十字最多与它有一个交点。如果在所有实数的集合上定义了一个函数(在R),然后每条垂直线与其图形相交于一点。鉴于属性 b) 和 c) 在我们的例子中,这个事实如何影响图的结构?哪一部分大号可以,哪些不能?我们的每个函数的图形都是一条多边形链(或直线),它“来自”北方或西北,在正方形内行进磷问吨小号,并在东北或东南方向消失。想象一个点,它出现在这个多边形链的左侧射线上并向右移动。那么这个点的速度矢量总是指向东北或东南方向,在多边形链的顶点处改变方向。
所有六个初始函数图都在想要的图中。所有其他图形都是由两条(左和右)射线组成的多边形链,可能还有来自正方形的几条线段磷问吨小号. 以下是其中一些多边形链(见图 4.13):ķ米问ñF,ķ米问吨和,ķ○ñĴ,Hñ吨和,HñĴ,在磷小号吨Ĵ,在米○F,在米○ñ吨和,在问吨和.

那么总共有多少条这样的多边形链呢?安排搜索的过程,找到足够的耐心和专注,我们最终可以推导出这种情况下的图形数量,其中线大号由六行构成。然而,这样的结果没有多大价值,因为解决这种特殊情况,最好记住它们的自然概括。

数学代写|组合学代写Combinatorics代考

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