数学 · 书 · English · 中文版 · 本站出品

Introductory Combinatorics

This course does not run through a list of formulas. It takes counting apart as an argument: every counting method comes from one act — building a bijection — and the rest are variations on it; each step is either a hypothesis of a theorem, something true by definition, or a modelling commitment asserting that a counting model represents something real; and every formula has conditions under which it gives a confident and wrong answer. The last chapter audits the first six.

结构交底

本课交付的结构

A counting problem, through being exchanged for a set in one-to-one correspondence with it that you already know how to count, turns from hard into solved.

It carries to test-case enumeration, shift scheduling, sampling design and probability modelling; it fails where what counts as 'the same object' has never been defined.

核心承诺

01Fix the equivalence first Before counting you must say what counts as the same object; change that and you have changed the problem.

02A bijection is a count Two sets in one-to-one correspondence are the same size; every counting technique is the construction of such a correspondence.

03Give the duplicates back Any scheme that counts an object more than once must say how many times, and that number must be the same for every object.

掌握路径

L1Restate and bound

学完能