##
**Enumerative combinatorics. Vol. 1.**
*(English)*
Zbl 0889.05001

Cambridge Studies in Advanced Mathematics. 49. Cambridge: Cambridge University Press. xi, 325 p. (1997).

[See Zbl 0608.05001 for the first edition published by Wadsworth.]

As the author explains in his introduction: “This book has three intended audiences and serves three different purposes. First, it may be used as a graduate-level introduction to a fascinating area of mathematics \(\dots\). The second intended audience consists of professional combinatorialists, for whom this book could serve as a general reference \(\dots\). Finally, this book may be used by mathematicians outside combinatorics whose work requires them to solve a combinatorial problem.”

The book serves all of these audiences well. The first chapter is a basic introduction to combinatorics and includes the fundamental counting formulas organized as counting functions under various conditions. The second chapter is devoted to a discussion of sieve methods. The remainder of the book consists of two long chapters: Partially ordered sets and Rational generating functions.

The book contains many careful examples and includes a large variety of exercises. The exercises are rated as to difficulty and a complete set of solutions is included. In addition each chapter contains a collection of historical notes and an extensive set of references.

As the author explains in his introduction: “This book has three intended audiences and serves three different purposes. First, it may be used as a graduate-level introduction to a fascinating area of mathematics \(\dots\). The second intended audience consists of professional combinatorialists, for whom this book could serve as a general reference \(\dots\). Finally, this book may be used by mathematicians outside combinatorics whose work requires them to solve a combinatorial problem.”

The book serves all of these audiences well. The first chapter is a basic introduction to combinatorics and includes the fundamental counting formulas organized as counting functions under various conditions. The second chapter is devoted to a discussion of sieve methods. The remainder of the book consists of two long chapters: Partially ordered sets and Rational generating functions.

The book contains many careful examples and includes a large variety of exercises. The exercises are rated as to difficulty and a complete set of solutions is included. In addition each chapter contains a collection of historical notes and an extensive set of references.

Reviewer: J.E.Graver (Syracuse)

### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05A15 | Exact enumeration problems, generating functions |

05A16 | Asymptotic enumeration |

06A07 | Combinatorics of partially ordered sets |

### Keywords:

enumerative combinatorics; partially ordered sets; generating functions; counting; sieve methods### Citations:

Zbl 0608.05001
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\textit{R. P. Stanley}, Enumerative combinatorics. Vol. 1. Cambridge: Cambridge University Press (1997; Zbl 0889.05001)

### Digital Library of Mathematical Functions:

§26.13 Permutations: Cycle Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis§26.14(ii) Generating Functions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis

§26.14(i) Definitions ‣ §26.14 Permutations: Order Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis

§26.15 Permutations: Matrix Notation ‣ Properties ‣ Chapter 26 Combinatorial Analysis

§26.17 The Twelvefold Way ‣ Properties ‣ Chapter 26 Combinatorial Analysis

Example 3 ‣ §26.18 Counting Techniques ‣ Properties ‣ Chapter 26 Combinatorial Analysis

Chapter 26 Combinatorial Analysis