What is Combinatorics
What is Combinatorics
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什么是组合学
DOI：
10.1017/cbo9780511803888.002
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
P. Cameron
中科院分区：
文献类型：
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作者：
P. Cameron
组合学是拓扑学的贫民窟。 J. H. C. Whitehead (attr.) 我不得不承认他在组合分析方面并不差——然而，即使在那时我也认为这个分支已经枯竭了。 Stanislaw Lem, His Master's Voice (1968) 组合学很特别。大多数可以在讲座课程中涵盖的数学主题都是为了一个单一的、明确的目标而构建的，例如柯西定理或素数定理。即使不存在这样一个明确的目标，也有一个明确的焦点（也许是有限群，或者非参数统计）。相比之下，组合数学似乎是随机选择的不相关谜题的集合。有两个因素促成了这一点。首先，组合学是广泛的而不是深刻的。它的触角几乎延伸到数学的各个角落。其次，重的是技术而不是结果。就像在一张网中一样，线程贯穿整个结构，出乎意料地出现在距离我们上次看到它们的地方很远的地方。忽视这一点的组合学处理必然会给人留下肤浅的印象。这一特点使教师的工作变得更加困难。阅读或讲授本质上是一维的。如果我们遵循一条线索，我们就会错过该主题本质上的相互联系。我尝试通过各种方法来解决这个困难。每一章都以该章中考虑的主题、技术和算法的列表以及对其他章节的交叉引用开始。另外，有些材料被设置为较小的类型，可以视为可选的。
Combinatorics is the slums of topology. J. H. C. Whitehead (attr.) I have to admit that he was not bad at combinatorial analysis — a branch, however, that even then I considered to be dried up. Stanislaw Lem, His Master's Voice (1968) Combinatorics is special. Most mathematical topics which can be covered in a lecture course build towards a single, well-defined goal, such as Cauchy's Theorem or the Prime Number Theorem. Even if such a clear goal doesn't exist, there is a sharp focus (finite groups, perhaps, or non-parametric statistics). By contrast, combinatorics appears to be a collection of unrelated puzzles chosen at random. Two factors contribute to this. First, combinatorics is broad rather than deep. Its tentacles stretch into virtually all corners of mathematics. Second, it is about techniques rather than results. As in a net, threads run through the entire construction, appearing unexpectedly far from where we last saw them. A treatment of combinatorics which neglects this is bound to give a superficial impression. This feature makes the teacher's job harder. Reading, or lecturing, is inherently one-dimensional. If we follow one thread, we miss the essential interconnectedness of the subject. I have attempted to meet this difficulty by various devices. Each chapter begins with a list of topics, techniques, and algorithms considered in the chapter, and cross-references to other chapters. Also, some of the material is set in smaller type and can be regarded as optional.