+- MKLab (https://mklab.gr)
+-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1)
+--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3)
+---- Forum: BOOKS (https://mklab.gr/forumdisplay.php?fid=6)
+----- Forum: NEW BOOKS (https://mklab.gr/forumdisplay.php?fid=42)
+------ Forum: FOREIGN (https://mklab.gr/forumdisplay.php?fid=91)
+------- Forum: PURE AND APPLIED MATHS (https://mklab.gr/forumdisplay.php?fid=94)
+-------- Forum: PROBABILITY&STATISTICS (https://mklab.gr/forumdisplay.php?fid=165)
+-------- Thread: Counting: The Art of Enumerative Combinatorics [Martin] (/showthread.php?tid=1683)
Counting: The Art of Enumerative Combinatorics [Martin] - mklabgr - 08-17-2026
Counting: The Art of Enumerative Combinatorics
Author: George E. Martin
Publication date: June 21, 2001
Publisher: Springer
Series:Undergraduate Texts in Mathematics
George E. Martin’s Counting: The Art of Enumerative Combinatorics is an accessible introduction to enumerative combinatorics—the mathematics of answering questions of the form “How many ways can this be done?” Rather than beginning with a heavy abstract framework, Martin develops the subject through concrete problems involving permutations, selections, arrangements, colors, objects, and symmetries. The book assumes essentially no formal prerequisites beyond mathematical maturity, making it appropriate for undergraduate mathematics, computer science, or statistics students and even strong secondary-school students. Its problem-oriented approach encourages readers to discover counting principles rather than merely memorize formulas.
The mathematical scope becomes progressively richer. Martin starts with elementary enumeration, including permutations, combinations, the binomial theorem, and familiar problems such as the birthday problem. He then develops the principle of inclusion–exclusion and generating functions, two fundamental tools for solving more complicated counting problems. The discussion subsequently moves into groups and group actions, including Burnside’s lemma, showing how symmetry can dramatically simplify enumeration. Later chapters cover recurrence relations, mathematical induction, and graph theory, giving the reader a surprisingly broad introduction to discrete mathematics within a relatively compact book.
A major strength of Counting is its emphasis on learning mathematics by solving problems. The text contains a very large collection of exercises—Chapter 1 alone reportedly contains 245 problems—and many are designed to make the reader experiment before the underlying principle is formally explained. Reviewers have particularly praised Martin's clear and engaging writing; Mathematical Reviews described it as genuinely suitable for undergraduate teaching, while The Mathematical Gazette highlighted how effectively Martin brings combinatorics to life. This makes the book useful not only as a textbook but also for self-study, mathematics teachers, problem-solving enthusiasts, and students preparing to study more advanced combinatorics.
Key Takeaways
- Counting is about structure, not just arithmetic: sophisticated enumeration problems become manageable once the correct representation or principle is identified.
- A strong toolkit is developed: permutations and combinations lead naturally to inclusion–exclusion, generating functions, recurrence relations, and Burnside's lemma.
- Problems drive the exposition: Martin emphasizes mathematical discovery through examples and exercises rather than presenting a catalogue of formulas.
- Excellent bridge to higher combinatorics: it starts at an approachable level but introduces ideas that lead naturally toward graph theory, algebraic methods, and more advanced discrete mathematics.