"Combinatorial Mathematics" 01-The Science of "placement"

In the middle school everyone has learned "permutation combination", computer department in the university also learned "discrete mathematics." Unlike the research objects (continuous variables) in mainstream mathematics, they mainly discuss the layout between discrete objects. These problems are ubiquitous and are found in a wide variety of puzzles and math games. They involve a relatively elementary mathematical knowledge, but need very strong skills and brainpower, the world's unresolved problems are abound. It is difficult to generalize the common points of these problems, and the solution itself does not form a unified theoretical system, often with a variety of loose methods and techniques.

Broadly speaking, there are many disciplines that can be counted as discrete objects, which can even include logics, algebra, and geometry, which have formed a large branch of the theoretical system. Therefore, it is unreasonable and meaningless to say discrete mathematics as a branch of mathematics. The "Discrete Mathematics" in the university curriculum is only a patchwork of some disciplines, it is a applied mathematics for the computer department, in order to compress the class, the material has been pressed and re-pressed. So what we've learned doesn't represent the whole discrete math, and the other chapters in the textbook, in addition to graph theory, are useless to students in computer science.

When it comes to computer mathematics, there are some discrete problems, they come from the needs of computers, or can be solved by computer, it is the existence of discrete object layout, counting, the optimal problem. This kind of problem is multifarious, seemingly simple but full of challenge, in which the graph theory is the typical representative. In mathematics, these complex problems are also integrated into a discipline, which is called Combinatorial mathematics (combinatorics, also called combinatorics or combinatorial theory), and with the development of computer, combinatorial mathematics has been widely valued and applied. Combinatorial theory is already the core theory of computer theory analysis and algorithm design, if you want to give computer science students to recommend a math course, I think combinatorial mathematics is more appropriate.

Combinatorial mathematics is born out of mathematical games or elementary problems, and therefore is not respected for a long time and is considered to be inadequate and lacking in systems. But in the second half of the 20th century, combinatorial theory is not only widely used in computer field, but also many ideas and conclusions are applied to the mainstream mathematics field. So far, combinatorial theory has become a branch of mainstream mathematics, although the problem-solving as the main work, but the process of the wonderful ideas and other mathematical theories have no merits and demerits. Many of the unresolved questions are also reminding people of how shallow and feeble our knowledge is, and that this ancient discipline is being re-presented in a brand new posture.

However, after years of development, the internal composition of mathematics has differentiated a lot of branches, an ordinary textbook is unable to cover its main content. Most of the textbooks are actually just a primer, and they don't even touch the basic conclusions of combinatorial mathematics. Just the introductory knowledge is already a very interesting and useful material for us, and the common methods and tools here can make you feel the beauty of mathematics. So this blog is just an introductory introduction to combinatorial mathematics, and deeper content (including graph theory) can only be left for later capture. Reference [] is a relatively complete combination of teaching materials, by the industry experts together cast, is a deep combination of the must-read classics.

The material selected here is no different from the general textbook, and most of the problems discussed are the counting problems of some discrete objects. I choose to first study combinatorial mathematics, in fact, mainly for the latter of the probability of paving the way, because the classical probability theory to a large extent first to solve the various counting problems. To the specific probability theory course, we can focus on the probability theory itself, rather than solve various elementary probability problems. But the counting is only a part of the combinatorial mathematics, and there are quite a lot of parts including combinatorial design, combinatorial optimization, graph theory, and so on, as with most textbooks, we choose to weaken the content and leave it to the later.

Catalog of this series

02-möbius Inversion formula

03-female functions and recursive relationships

04-Basic Counting issues

05-Classic Counting method

06-Combination Design

　　The total list of blogs is here

"Pre-order Discipline" set theory, abstract algebra, linear algebra, calculus

Resources

[1] Lecture Notes on combinatorial mathematics, Liqiao, 2008

A very good domestic textbooks, but also the main reference for this blog. It contains not only the knowledge of elementary combinatorial theory, but also the extension and content of a considerable amount of space. The background of the concept and the author's understanding of Combinatorial learning are also expounded. Classic books and articles are recommended in the appendix, which is quite enlightening.

[2] Combinatorial Mathematics (introductory combinatorics,5th), R.A. brualdi,2012

This should be the most widely used textbook in combinatorial mathematics, covering all the fundamentals of combinatorial learning. The problem-centric, the knowledge of the series together. It is perfect for self-study. A lot of material and examples of exercise, are worth thinking over and over again.

[3] Combinatorial mathematics, Cao Rucheng, 2000

[4] Combinatorial Mathematics (3rd), Lukaishing, 2002

These two are also very excellent textbooks, in the selection, layout and even the use of symbols are already very mature, from the two began self-study is also a good choice.

[5] Combinatorial theory (upper and lower), Koering, 1981

A relatively old textbook, but the content has been very rich and comprehensive. The author tries to express completely and rigorously all the questions, which is a more suitable teaching material for self-study.

[6] Combinatorial mathematics, H.J. reise,1983

A simple reading of combinatorial mathematics, a short Baishi page covers all the basics. Simple and easy to understand, well-organized, suitable for self-study. Each chapter also has the problem of traction, and lists a large number of expanded reading, can be used as a guide to combinatorial theory.

[7] Combinatorial Mathematics theory and solving, Wang Yuanyuan, 1989

A very good exercise to explain, which only involves combinatorial mathematics the most basic knowledge, without too deep theory. On the one hand, the exercises can help to understand the thought and conclusion of combinatorial theory, on the other hand, it is also the thinking exercise and summary.

"Combinatorial Mathematics" 01-The Science of "placement"