Discrete Mathematics: A Combinatorial Approach [Athanasiadis]

[mklabgr]

Discrete Mathematics: A Combinatorial Approach
Author: Christos A. Athanasiadis
Publication date: 16 February 2026 (eBook); 17 February 2026 (hardcover)
Publisher: Springer Cham
Series:Undergraduate Texts in Mathematics
Pages: XI + 329
DOI: 10.1007/978-3-032-14290-0 (Springer)

Discrete Mathematics: A Combinatorial Approach is an introductory university-level textbook on discrete mathematics with a strong emphasis on combinatorics. It is intended primarily for undergraduate students in mathematics and computer science, but it is also suitable for independent learners and mathematical problem solvers who already have some familiarity with proofs. The prerequisites are deliberately modest. Rather than presenting discrete mathematics as a collection of unrelated techniques, Christos A. Athanasiadis develops combinatorial reasoning as the central theme, frequently illustrating the same result through different proofs to show how mathematical ideas connect.

The book begins with basic counting principles and enumerative combinatorics, then develops the inclusion–exclusion principle, set partitions, equivalence relations and partially ordered sets. A substantial chapter introduces graph theory, including topics such as graph coloring, matching and Kruskal's algorithm. Particular attention is then given to generating functions, recurrence relations and combinatorial identities—one of the book's distinguishing emphases. The final main chapter introduces discrete probability, demonstrating both how combinatorial techniques can solve probability problems and how probabilistic reasoning can illuminate combinatorial questions.

A major strength is its problem-oriented presentation. Examples appear early and frequently, and every chapter contains exercises ranging from straightforward applications to substantially more challenging problems. Hints and solutions to selected exercises are collected at the end. This makes the book particularly suitable for a one-semester discrete mathematics course, but also useful as preparation for deeper study of combinatorics, graph theory, probability, algorithms and theoretical computer science. The author, Christos A. Athanasiadis, is Professor of Mathematics at the National and Kapodistrian University of Athens; his research specializes in algebraic and geometric combinatorics. 

Main topics

• Elementary enumerative combinatorics
  
• Inclusion–exclusion
  
• Pigeonhole principle
  
• Set partitions and equivalence relations
  
• Partial orders
  
• Graph theory
  
• Graph coloring and matchings
  
• Generating functions
  
• Linear recurrence relations
  
• Combinatorial identities
  
• Discrete probability and random variables 
  

Key takeaways
1. Combinatorics is the core perspective. The book teaches discrete mathematics primarily by developing ways of counting, structuring and reasoning about finite objects.
2. Generating functions receive unusual prominence. They are treated as a fundamental tool rather than an optional advanced topic, making the book particularly useful for students interested in combinatorics.
3. It emphasizes mathematical thinking rather than formulas. Multiple proofs, abundant examples and exercises are used to demonstrate connections between apparently different discrete structures.
4. Best suited to: undergraduate mathematics/computer-science students, mathematics teachers, competition-oriented students, and anyone wanting a rigorous but accessible bridge into combinatorics and graph theory. 

BOOK