Inquiry-Based Enumerative Combinatorics: One, Two, Skip a Few... Ninety-Nine, One Hundred

Description

This textbook offers the opportunity to create a uniquely engaging combinatorics classroom by embracing Inquiry-Based Learning (IBL) techniques. Readers are provided with a carefully chosen progression of theorems to prove and problems to actively solve. Students will feel a sense of accomplishment as their collective inquiry traces a path from the basics to important generating function techniques. Beginning with an exploration of permutations and combinations that culminates in the Binomial Theorem, the text goes on to guide the study of ordinary and exponential generating functions. These tools underpin the in-depth study of Eulerian, Catalan, and Narayana numbers that follows, and a selection of advanced topics that includes applications to probability and number theory. Throughout, the theory unfolds via over 150 carefully selected problems for students to solve, many of which connect to state-of-the-art research. Inquiry-Based Enumerative Combinatorics is ideal for lower-division undergraduate students majoring in math or computer science, as there are no formal mathematics prerequisites. Because it includes many connections to recent research, students of any level who are interested in combinatorics will also find this a valuable resource. Front Matter ....Pages i-xi Introduction to this book (T. Kyle Petersen)....Pages 1-14 First principles (T. Kyle Petersen)....Pages 15-31 Permutations (T. Kyle Petersen)....Pages 33-41 Combinations (T. Kyle Petersen)....Pages 43-53 The Binomial Theorem (T. Kyle Petersen)....Pages 55-64 Recurrences (T. Kyle Petersen)....Pages 65-76 Generating functions (T. Kyle Petersen)....Pages 77-88 Exponential generating functions and Bell numbers (T. Kyle Petersen)....Pages 89-100 Eulerian numbers (T. Kyle Petersen)....Pages 101-111 Catalan and Narayana numbers (T. Kyle Petersen)....Pages 113-128 Refined Enumeration (T. Kyle Petersen)....Pages 129-146 Applications to probability (T. Kyle Petersen)....Pages 147-163 Some partition theory (T. Kyle Petersen)....Pages 165-177 A bit of number theory (T. Kyle Petersen)....Pages 179-196 Back Matter ....Pages 197-238