1: What is Combinatorics?
- Page ID
- 71240
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- 1.1: Enumeration
- Enumeration is a big fancy word for counting. If you’ve taken a course in probability and statistics, you’ve already covered some of the techniques and problems we’ll be covering in this course. When a statistician (or other mathematician) is calculating the “probability” of a particular outcome in circumstances where all outcomes are equally likely, what they usually do is enumerate all possible outcomes, and then figure out how many of these include the outcome they are looking for.
- 1.2: Graph Theory
- When a mathematician talks about graph theory, she is not referring to the “graphs” that you learn about in school, that can be produced by a spreadsheet or a graphing calculator. The “graphs” that are studied in graph theory are models of networks. Any network can be modeled by using dots to represent the nodes of the network (the cities, computers, cell phones, or whatever is being connected) together with lines to represent the connections between those nodes. This model is called a graph.
- 1.3: Ramsey Theory
- Ramsey theory takes its name from Frank P. Ramsey, a British mathematician who died in 1930 at the tragically young age of 26, when he developed jaundice after an operation. Ramsey was a logician. A result that he considered a minor lemma in one of his logic papers now bears the name “Ramsey’s Theorem” and was the basis for this branch of mathematics.
- 1.4: Design Theory
- Like graph theory, design theory is probably not what any non-mathematician might expect from its name. When researchers conduct an experiment, errors can be introduced by many factors (including, for example, the timing or the subject of the experiment). It is therefore important to replicate the experiment a number of times, to ensure that these unintended variations do not account for the success of a particular treatment.
- 1.5: Coding Theory
- Coding theory is the study of encoding information into different symbols. When someone uses a code in an attempt to make a message that only certain other people can read, this becomes cryptography. In coding theory, we ignore the question of who has access to the code and how secret it may be. Instead, one of the primary concerns becomes our ability to detect and correct errors in the code.
- 1.6: Summary
- This page contains the summary of the topics covered in this chapter.