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.
T. Kyle Petersen is Professor of Mathematics at DePaul University in Chicago. His research interests lie in algebraic, enumerative, and topological combinatorics, and he has been an active member of the Inquiry-Based Learning (IBL) community for over a decade. His graduate textbook, Eulerian Numbers, appears in Birkhäuser Advanced Texts Basler Lehrbücher.
Author(s): Petersen, T. Kyle
Series: Undergraduate Texts in Mathematics
Edition: 1
Publisher: Springer International Publishing
Year: 2019
Language: English
Pages: XI, 238
Preface......Page 6
Contents......Page 10
Image Credits......Page 12
An introduction to combinatorial problem solving......Page 13
An inquiry based approach......Page 19
The toolbox......Page 21
Structure of this book......Page 23
A word about proofs......Page 24
1 First principles......Page 27
Organizing data......Page 38
2 Permutations......Page 44
The symmetric group......Page 49
3 Combinations......Page 53
Anagrams, or multiset permutations......Page 59
4 The Binomial Theorem......Page 64
Counting permutations according to cycles......Page 68
5 Recurrences......Page 74
Lucas numbers and polynomials......Page 80
6 Generating functions......Page 86
Money problems......Page 92
7 Exponential generating functions and Bell numbers......Page 98
OEIS......Page 107
8 Eulerian numbers......Page 110
Euler (not Eulerian) numbers......Page 116
9 Catalan and Narayana numbers......Page 121
Weak order and the Tamari lattice......Page 129
10 Refined Enumeration......Page 137
Continued fractions......Page 146
11 Applications to probability......Page 155
Longest increasing subsequences and random partitions......Page 163
12 Some partition theory......Page 172
Plane Partitions......Page 178
13 A bit of number theory......Page 185
General Möbius Inversion......Page 196
Appendix: Supplementary exercises......Page 203
