This book offers a systematic and accessible introduction to enumerative combinatorics, the branch of mathematics concerned with counting discrete structures. Each topic is developed from first principles and supported by carefully chosen examples and exercises.

The first part establishes the foundations of combinatorial reasoning. It introduces the basic counting principles and their applications to classical sequences such as binomial coefficients, Stirling numbers, and Bell numbers, and provides entry points to discrete probability and graph theory, both as tools and as sources of enumerative problems. The second part presents the central methods of modern enumeration. Recurrence relations and generating functions are developed as primary techniques, with applications to integer partitions, permutations, and related structures. The final chapter turns to unlabelled structures, introducing group actions as a framework for understanding symmetry in counting.

Aimed at advanced undergraduate and beginning graduate courses, the clear exposition and numerous exercises make the book suitable for self-study.

Bridges elementary counting and advanced methods, offering a clear pathway into enumerative combinatorics Integrates theory with graded exercises, suitable for classroom use as well as self-study Explains generating functions with clear combinatorial interpretations

Autorentext

Alexander Omelchenko is a professor of applied mathematics at Constructor University, Bremen, Germany. His teaching and writing focus on proof oriented courses in graph theory, discrete mathematics, and enumerative combinatorics. His research spans graph theory and enumerative combinatorics as well as mathematical physics and applied topology, with contributions to the enumeration of maps on surfaces and related topics. Over his career, he has designed and delivered undergraduate and graduate curricula in mathematics and computer science and developed widely used online courses in discrete mathematics and graph theory.

Inhalt

Chapter 1. Elementary Combinatorics.- Chapter 2. Elements of Discrete Probability.- Chapter 3. Basic Graph Theory.- Chapter 4. Recurrences and Generating Functions.- Chapter 5. Generating Functions and Labeled Enumeration.- Chapter 6. Enumeration of Unlabeled Objects.