# 数学代写|组合学代写Combinatorics代考|MATH393

## 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.

# 组合学代写

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

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

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

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?

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