We present here the first of ten units of study used by the author to train beginning graduate students and advanced undergraduates for further study in the general topic of combinatorial algorithms. This unit of study focuses on mathematical concepts used to linearly order sets of objects. Our general approach is to focus on the "geometric" theory of algorithms by which we mean the use of graphical or pictorial ways to understand what is going on. This approach is both fun and powerful, preparing you to invent your own algorithms for a wide range of problems.

The beautiful mathematical counting theory due originally to George Polya is developed and extended in this unit of study. We discuss many extensions of the classical results of Polya and address problems of actually constructing systems of distinct representatives under the constraints of symmetry. We focus on mathematical concepts as well as on the vizualization of algorithms through the use of graphical and pictorial methods.

These units of study present the basics of sorting and listing from the point of view of combinatorial algorithms. We focus on mathematical concepts as well as on the vizualization of algorithms through the use of graphical and pictorial methods. Sorting and merging algorithms are discussed, including sorting networks. We study listing, ranking and unranking algorithms for the basic sets that serve as building blocks of combinatorics. This approach will give you the tools to invent your own algorithms for a wide range of problems.

This book is based on Professor Williamson's twenty-six years of teaching calculus at the University of California, San Diego. It is a revised and updated version of a "tutors' guide" that he handed out to students wanting to tutor for his classes in integral calculus. Mostly, these tutors were a great help. But when they made mistakes in explaining technique or concept, these mistakes were hard to detect and rectify before the final exam. Tutoring Integral Calculus covers and hopefully rectifies the most common sources of tutoring difficulties.

Discrete mathematics is fundamental to computer science, and this up-to-date text assists undergraduates in mastering the ideas and mathematical language to address problems that arise in the field's many applications. It consists of 4 units of study: counting and listing, functions, decision trees and recursion, and basic concepts of graph theory.