1.1. From the Abstract.

A general theory is developed for the enumeration of order-reversing maps of finite ordered sets P into chains. This theory encompasses many apparently disparate topics in combinatorial theory, including (1) ordinary partitions, (2) ordered partitions (compositions), (3) plane and multidimensional partitions, with applications to Young tableaux, (4) the Eulerian numbers and their refinements, (5) the tangent and secant numbers (or Euler numbers) and their refinements, (6) the indices of permutations, (7) trees, (8) stacks, and (9) protruded partitions, with applications to the Fibonacci numbers. The main tool used is that of generating functions. In particular, we study how the structure of $P$ influences the form of the generating functions under consideration. As an application, we derive new combinatorial relationships between a finite ordered set $P$ and its distributive lattice of order ideals.

1.2. Review by: Lincoln K Durst.
Mathematical Reviews MR0332509 (48 #10836).

Abstract as above is quoted.

2.1. From the Preface.

These notes are based on a series of eight lectures given at the University of Stockholm during April and May, 1981. They were intended to give an overview of two topics from "combinatorial commutative algebra", viz., (1) solutions to linear equations in nonnegative integers (which is equivalent to the theory of invariants of a torus acting linearly on a polynomial ring), and (2) the face ring of a simplicial complex. In order to give a broad perspective many details and specialised topics have been regretfully omitted. In general, proofs have been provided only for those results which were obscure or inaccessible in the literature at the time of the lectures. The original lectures presupposed considerable background in commutative algebra, homological algebra, and algebraic topology. In order to broaden the accessibility of these notes, Chapter 0 has been prepared with the kind assistance of Karen Collins. This chapter provides most of the background information in algebra, combinatorics, and topology needed to read the subsequent chapters.

2.2. From the Abstract.

The purpose of this introduction is to provide the reader with the relevant background from combinatorics, algebra, and topology for understanding of the text. In general the reader may prefer to begin with Chapter I and refer back to this chapter only when necessary. We assume the reader is familiar with standard (first-year graduate) material but has no specialised knowledge of combinatorics, commutative algebra, homological algebra, or algebraic topology.

2.3. Review by: Peter McMullen.
Mathematical Reviews MR0725505 (85b:05002).

Over the past decade or so, commutative algebra has played an increasingly important role in the investigation of certain classes of combinatorial problems, and the present author has been in the forefront of those working in the field. In this book, he gives an overview of two of the main areas involved - solutions of linear equations in nonnegative integers, and numbers of faces of simplicial complexes. Chapter 0 is devoted to background material on combinatorics, algebra and topology; all that is subsequently required is here, although the treatment is very condensed. ... In Chapter 1, the application is to the enumeration of integer stochastic matrices (magic squares), volumes of rational convex polytopes, and certain combinatorial reciprocity theorems. In Chapter 2 the Stanley-Reisner ring of a simplicial complex is considered, leading to a proof of the upper bound theorem for spheres. A number of previously unpublished results are also included in the course of the discussion. Though this is a comparatively slim volume (even, unfortunately, lacking an index), it contains a wealth of material, and is surely indispensable reading for anyone interested in recent developments in combinatorics.

3.1. Review by: Joe Roberts.
Mathematical Reviews MR0847717 (87j:05003).

In a little over 300 pages of careful exposition, the author has packed a tremendous amount of information into this book. This has been possible partly due to the fact that over one third of the book is devoted to 180 graded exercises (many with multiple parts) and their solutions (or references to solutions). The book is very up to date, is very detailed, and quite general without seeking the greatest generality possible. The book demands of the reader considerable mathematical maturity as well as considerable familiarity with undergraduate mathematics. The chapter headings are: What is enumerative combinatorics? (63 pages), Sieve methods (32 pages), Partially ordered sets (106 pages), Rational generating functions (91 pages). In addition there are notes and references to sources, with 122 citations in all. In his introduction the author states that his intentions are as follows: to provide a graduate level introduction to enumerative combinatorics, to give a general reference to professionals in the field, and to provide a book useful to other mathematicians needing to solve a combinatorial problem. In the reviewer's opinion the author has succeeded admirably in realising these objectives. He has produced an excellent and valuable book on the subject. We look forward to seeing the second volume.

3.2. Review by: Earl Glen Whitehead, Jr.
SIAM Review 30 (1) (1988), 170-171.

In the Foreword to this book, G-C Rota writes, "It has been said that combinatorics has too many theorems, matched with very few theories; Stanley's book belies this assertion." I agree with Rota that a series of theorems proven by ad hoc methods is a poor way to present combinatorics. On the other hand, too much theory without sufficient practical applications is also a poor way to present combinatorics. In my opinion, Stanley puts too much stress on theory in this book.

One obvious omission is that Stanley failed to give adequate coverage to Pólya's enumeration theorem. Pólya's theory is undoubtedly one of the most useful theories in enumerative combinatorics. It is based on the theory of permutation groups. In this book Stanley stressed the theory of partial ordered sets at the expense of other theories useful in enumerative combinatorics.
...
In the Preface, Stanley writes that this book is intended for three audiences. (1) This book may be used to introduce graduate students to enumerative combinatorics. (2) This book may be used as a general reference by professional combinatorialists. (3) This book may be used by other mathematicians who need to solve a combinatorial problem. ... I agree that Stanley's book is appropriate as a reference for professional combinatorialists. I doubt that many other mathematicians will find this book useful in solving combinatorial problems that occur in their work.

4.1. From the Publisher.

Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists.

New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors.

4.2. Review by: Volkmar Welker.
Mathematical Reviews MR1453579 (98h:05001).

In his review of the first edition of this book, Peter McMullen writes: "Though this is a comparatively slim volume (even, unfortunately, lacking an index), it contains a wealth of material, and is surely indispensable reading for anyone interested in recent developments in combinatorics." Now the second edition is out; it is about double the size of the first edition, and there is an index. The first two chapters have been thoroughly revised and updated (so McMullen's review still covers their content), and a substantial third chapter has been added that covers new material. ... But during all the years that the first edition has been the indispensable tool, as McMullen had predicted, the field of combinatorial commutative algebra has grown so much that again the book can only give an introduction to the topics that have developed and are developing. Therefore, the author confines himself in the new Chapter III to the very fruitful interaction between the combinatorics and commutative algebra of the face ring or Stanley-Reisner ring of a simplicial complex. The chapter starts with a section on simplicial polytopes, toric varieties and the g-theorem. Then the commutative algebra of shellable simplicial complexes is discussed. In particular, Stanley gives a definition of a sequentially Cohen-Macaulay complex, a concept that is the algebraic counterpart to non-pure shellability as introduced by A Björner and M L Wachs [(1996)]. Then, in two sections special classes of simplicial complexes (e.g. matroid complexes, balanced complexes) are studied and flag f- and h-vectors are introduced. The next section deals with modules of spline functions on spaces triangulated by a given simplicial complex. A section on algebras with straightening laws and one on relative simplicial complexes follow. After presenting results on the implication of free group actions on the commutative algebra of a simplicial complex, in two final sections the algebraic and combinatorial aspects of passing to subcomplexes and subdivisions are covered. Then the chapter closes with a wealth of solved and unsolved problems on the face ring of a simplicial complex.

As mentioned before, the developments in the field make it impossible to cover all of the material. Notably, the combinatorial and geometric theory of Gröbner bases is only touched upon. Here B Sturmfels's book [Gröbner bases and convex polytopes] provides an excellent introduction. Also, for a more elementary introduction to some of the material in the book under review one can now consult the book by T Hibi [Algebraic combinatorics on convex polytopes]. There is also a chapter on combinatorial aspects of Cohen-Macaulay rings in the book by W Bruns and J Herzog [Cohen-Macaulay rings]. This book is Stanley's major source for facts from commutative algebra.

Let me close the review by echoing McMullen: the book is an indispensable tool for a mathematician working on the borderline between combinatorics and commutative algebra.

5.1. From the Publisher.

This book is the first of a two-volume basic introduction to enumerative combinatorics at a level suitable for graduate students and research mathematicians. It concentrates on the theory and application of generating functions, a fundamental tool in enumerative combinatorics. The book covers those parts of enumerative combinatorics of greatest applicability to other areas of mathematics. The four chapters are devoted to an introduction to enumeration (suitable for advanced undergraduates), sieve methods (including the Principle of Inclusion-Exclusion), partially ordered sets, and rational generating functions. There are a large number of exercises, almost all with solutions, which greatly augment the text and provide entry into many areas not covered directly. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.

5.2. Review by: Wayne M Dymacek.
Mathematical Reviews MR1442260 (98a:05001).

Much of the review for the original version of this book [R P Stanley, Enumerative combinatorics. Vol. I, 1986] is still true. This revision still provides "a graduate level introduction to enumerative combinatorics", still gives "a general reference to professionals in the field", and still is "useful to other mathematicians needing to solve a combinatorial problem''. In addition, there are 65 new exercises for a total of 245 and over 75 corrections or additions which keep the book up to date.

6.1. From the Publisher.

This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and representation theory. An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.

6.2. Review by: Ira M Gessel.
Mathematical Reviews MR1676282 (2000k:05026).

Volume 1 of Stanley's Enumerative combinatorics appeared in 1986 and since then its readers have eagerly awaited Volume 2. They will not be disappointed. Volume 2 not only lives up to the high standards set by Volume 1, but surpasses them. The text gives an excellent account of the basic topics of enumerative combinatorics not covered in Volume 1, and the exercises cover an enormous amount of additional material.

Volume 2 contains only three chapters, numbered 5, 6, and 7, but they contain much more than the four chapters of Volume 1. Chapter 5 begins with an account of exponential generating functions ... Chapter 6 deals with algebraic, D-finite, and noncommutative generating functions. Rational power series were discussed in Volume 1, and algebraic power series are an important generalisation. The prototypical algebraic function that arises in enumeration is the generating function for Catalan numbers ... Chapter 7, by far the longest of the three chapters, is on symmetric functions. These "symmetric functions" are actually symmetric polynomials or formal power series, often in infinitely many variables. Although the theory of symmetric functions has several connections with combinatorics, the one that Stanley emphasises is with plane partitions. ...
...
As in Volume 1, there are historical notes at the end of each chapter, and these demonstrate Stanley's uniquely broad knowledge of the field. The numerous exercises go far beyond the text in introducing a large number of related topics. Many exercises contain material appearing here for the first time, and each exercise has either a solution or a reference to the literature. One highlight is a collection of 66 combinatorial interpretations for the Catalan numbers, including many that are little-known. The notes, exercises, and references together give an impressively thorough account of what is known about the field. ...

Stanley's book is a valuable contribution to enumerative combinatorics. Beginners will find it an accessible introduction to the subject, and experts will still find much to learn from it.

6.3. Review by: Ira M Gessel.
Bulletin of the American Mathematical Society 39 (1) (2002), 129-135.

Richard Stanley's Enumerative Combinatorics, Volume 1, appeared in 1986, and its interesting choice of topics, clear prose style, numerous exercises, and scholarly erudition quickly won it a large following. The long-awaited Volume 2 appeared in 1999, and it is an impressive work of scholarship that surpasses the high standards set by Volume 1. Stanley was the awarded the American Mathematical Society's 2001 Leroy P Steele Prize for Mathematical Exposition in recognition of the completion of the two volumes.
...
The two volumes of Enumerative Combinatorics cover nearly every major topic in enumerative combinatorics to a greater or lesser degree. The most important topic I can think of that is not mentioned at all is the enumeration of planar maps. (A good account can be found in Goulden and Jackson [Combinatorial Enumeration]; they also treat sequences and lattice paths more extensively than does Stanley.) A topic that receives only brief coverage is that of enumeration under group action, and in particular the enumeration of unlabelled graphs of various types. A comprehensive account of the theory up to 1972 was given by Harary and Palmer [Graphical Enumeration], and a more modern approach to the subject has been given recently by Bergeron, Labelle, and Leroux [Combinatorial Species and Tree-like Structures]. However, an exposition in which symmetric functions appear in the central role that (in this reviewer's opinion) they deserve does not yet exist.

Enumerative Combinatorics can be read on two levels. The text, together with the easier exercises, is a very thorough introduction to enumerative combinatorics that is accessible to a beginning graduate student or even a good undergraduate. With its more advanced exercises, historical notes, and extensive references, the books are an indispensable resource for the expert.

7.1. From the Introduction.

A recent newcomer to the centre stage of modern mathematics is the area called combinatorics. Although combinatorial mathematics has been pursued since time immemorial, and at a reasonable scientific level at least since Leonhard Euler (1707-1783), the subject has come into its own only in the last few decades. The reasons for the spectacular growth of combinatorics come both from within mathematics itself and from the outside.

Beginning with the outside influences, it can be said that the recent development of combinatorics is somewhat of a Cinderella story. It used to be looked down on by "mainstream" mathematicians as being somehow less respectable than other areas, in spite of many services rendered to both pure and applied mathematics. Then along came the prince of computer science with its many mathematical problems and needs - and it was combinatorics that best fitted the glass slipper held out.

The developments within mathematics that have contributed to the current strong standing of combinatorics are more difficult to pinpoint. One is that, after an era where the fashion in mathematics was to seek generality and abstraction, there is now much appreciation of and emphasis on the concrete and "hard" problems. Another is that it has been gradually more and more realised that combinatorics has all sorts of deep connections with the mainstream areas of mathematics, such as (to name the most important ones) algebra, geometry, probability and topology.

Our aim with this monograph is to give the reader some answers to the questions "What is combinatorics, and what is it good for?" We will do that not by attempting any kind of general survey, but by describing a few selected problems and results in some detail. We want to bring you some examples of problems from "pure" combinatorics, some examples illustrating its interactions with other parts of mathematics, and a few glimpses of its use for computer science. Fortunately, the problems and results of combinatorics are usually quite easy to state and explain, even to the layman with a solid knowledge of high school mathematics. Its accessibility is one of its many appealing aspects. For instance, most popular mathematical puzzles and games, such as Rubik's cube and jigsaw puzzles, are essentially problems in combinatorics.

To achieve our stated purpose it has been necessary to concentrate on a few topics, leaving many of the specialities within combinatorics without mention. The choice will naturally reflect our own interests. The discussion in the Notes section points to some more general accounts that can help remedy this shortcoming.

With some simplification, combinatorics can be said to be the mathematics of the finite. One of the most basic properties of a finite collection of objects is its number of elements. For instance, take words formed from the letters $a, b,$ and $c$, using each letter exactly once. There are six such words:

          $abc, acb, bac, bca, cab, cba.$

Now, say that we have $n$ distinct letters. How many words can be formed? The answer is $n.(n−1).(n−2) ... 3.2.1$, because the first letter can be chosen in $n$ ways, then the second one in $n-1$ ways (since the letter already chosen as the first letter is no longer available), the third one in $n-2$ ways, and so on. Furthermore, the total number must be the product of the number of individual choices.

The number of words that can be formed with $n$ letters is an example of an enumerative problem. Enumeration is one of the most basic and important aspects of combinatorics. In many branches of mathematics and its applications you need to know the number of ways of doing something.

One of the classical problems of enumerative combinatorics is to count partitions of various kinds, meaning the number of ways to break an object into smaller objects of the same kind. Various kinds of partitions - of numbers, of sets, and of geometric objects - are considered. In fact, the idea of partition can be said to be a leading theme in this book.

The study of partition enumeration was begun by Euler and is very active to this day. We will exposit some parts of this theory. All along the way there are interesting connections with algebra, but these are unfortunately too sophisticated to go into details here. We will, however, give a few hints of this connection, especially in Chapter 5. We also illustrate (in Chapter 15) the relevance of partitions to applied problems.

Another, more recent, topic within enumeration is to count the number of tilings. These are partitions of a geometric region into smaller regions of some specified kinds. We will give some glimpses of recent progress in this area. The mathematical roots are in this case mainly from statistical mechanics.

In Chapter 12 we present some progress made in the combinatorial study of convex polytopes. In three dimensions these are the decorative solid bodies with flat polygon sides (such as pyramids, cubes and geodesic domes) that have charmed and intrigued mathematicians and laymen alike since antiquity. In higher dimensions they can be perceived only via mathematical tools, but they are just as beautiful and fascinating. Of this huge subject we discuss the question of laws governing the numbers of faces of various dimensions on the boundary of a polytope.

Combinatorics is used in many ways in computer science, for instance for the construction and analysis of various algorithms. (Remark: algorithms are the logically structured systems of commands that instruct computers how to perform prescribed tasks.) Of this young but already huge and rapidly growing area we will give here but the smallest glimpse, namely a couple of examples from complexity theory. This is the part of theoretical computer science that concerns itself with questions about computer calculations of the type "How hard is it ?", "How much time will it take ?" Proving that you cannot do better than what presently known methods allow is often the hardest part, and the part where the most mathematics is needed. Our examples are of this kind.

To illustrate the surprising connections that exist between combinatorics and seemingly unrelated parts of mathematics we have chosen the links with topology. This is an area which on first acquaintance seems far removed from combinatorics, having to do with very general infinite spaces. Nevertheless, the tools of algebraic topology have proven to be of use for solving some problems from combinatorics and theoretical computer science. Again, the theme of enumeration in its various forms pervades some of this border territory.

Understanding this book should for the most part require no more than some basic knowledge of mathematical notation and concepts involving sets, functions, etc., such as taught in a course on precalculus. Some parts should be accessible to readers with even less background knowledge while others are more demanding, at least in some of the details. Generally speaking, we start out at a very elementary level and assume more and more mathematical background as we go along. We hope that this way the book is informative for laymen as well as for students and colleagues from other parts of mathematics.

7.2. Review by: Michael Weiss.
Mathematical Reviews MR2768279 (2012e:05002).

A combinatorial miscellany sets out to introduce readers to some easily-stated (but surprisingly subtle) problems in pure combinatorics, and to reveal the connections between this branch of mathematics and others, such as representation theory and algebraic topology. The authors have written a book that is intended for a mathematically sophisticated novice; that is, for a reader who may know little more than "a layman with a solid knowledge of high school mathematics" (p. 6), but who is nevertheless capable of following quite subtle and complicated arguments stated in technical notation. (The existence of such readers is a longstanding open conjecture in mathematics; despite decades of work, an existence proof remains elusive.) Notwithstanding the challenge of writing a text that is mathematically rich but at the same time accessible to novices, the authors have done a remarkable job. The book is eminently readable, at times almost conversational in tone, without ever seeming patronising or condescending.
...
...this book covers an extraordinarily wide-ranging set of topics. What is most impressive about the book is how the authors reveal connections between topics that superficially appear quite unrelated. Indeed perhaps the only serious flaw in this book is its title: referring to it as a "miscellany" suggests a somewhat disconnected set of self-contained topics, when in fact the authors have succeeded in producing a tightly interconnected, cohesive account of the many ways in which combinatorial techniques can be applied in a broad range of mathematical contexts.

8.1. From the Preface.

Enumerative combinatorics has undergone enormous development since the publication of the first edition of this book in 1986. It has become more clear what the essential topics are, and many interesting new ancillary results have been discovered. This second edition is an attempt to bring the coverage of the first edition more up to date and to impart a wide variety of additional applications and examples.

The main difference between this edition and the first is the addition of ten new sections (six in Chapter 1 and four in Chapter 3) and more than 350 new exercises. In response to complaints about the difficulty of assigning homework problems whose solutions are included, I have added some relatively easy exercises without solutions, marked by an asterisk. There are also a few organisational changes, the most notable being the transfer of the section to the theory of $(P, \omega)$-partitions from Chapter 4 to Chapter 3, and extending this section to the theory of $(P, \omega)$-partitions for any labelling ω. In addition, the old Section 4.6 has been split into Sections 4.5 and 4.6.

There will be no second edition of volume 2 nor a volume 3. Since the references in volume 2 to information in volume 1 are no longer valid for this second edition, I have included a table entitled "First Edition Numbering" which gives the conversion between the two editions for all numbered results (theorems, examples, exercises, etc., but not equations).

Exercise 4.12 has some sentimental meaning for me. This result, and related results connected to other linear recurrences with constant coefficients, is a product of my earliest research, done around the age of 17 when I was a student at Savannah High School.

"I have written my work, not as an essay which is to win the applause of the moment, but as a possession for all time."

It is ridiculous to compare Enumerative Combinatorics with History of the Peloponnesian War, but I can appreciate the sentiment of Thucydides. I hope this book will bring enjoyment to many future generations of mathematicians and aspiring mathematicians as they are exposed to the beauties and pleasures of enumerative combinatorics.

8.2. Review by: Darren Glass.
Mathematical Association of America (25 February 2012).
https://maa.org/press/maa-reviews/enumerative-combinatorics-vol-i

Some books seem like they shouldn't need reviews: The Bible, Moby Dick, Great Expectations, Euclid's Elements. Their names are so ingrained in our minds as the pinnacle of what a great book should be that it almost doesn't occur to us that a review is a good idea. Who is going to decide whether to read Shakespeare based on what folks on Amazon think of it? Now, Richard Stanley's Enumerative Combinatorics, Volume I is probably not quite at the level of the above-named classics, but it is about as close as a graduate-level text in mathematics can be. Don't believe me? Look at what others said about the first edition:

"In a little over 300 pages of careful exposition, the author has packed a tremendous amount of information into this book." - MathSciNet

"This is a masterful work of scholarship which is, at the same time, eminently readable and teachable. It will be the standard work in the field for years to come." - Citation for 2001 Leroy Steele Prize For Mathematical Exposition, which Stanley won for the two volumes of his book.

"Historically then this is a book of major importance. It provides a widely accessible introduction to many topics in combinatorics ... Furthermore, it is sure to become a standard as an introductory graduate text in combinatorics." - George Andrews, writing in the Bulletin of the American Mathematical Society.

"Best of all, Stanley has succeeded in dramatising the subject, in a book that will engage from start to finish the attention of any mathematician who will open it at page one." - Gian-Carlo Rota, in the preface to the first edition.

Stanley has recently revised the first volume of Enumerative Combinatorics, citing the words of Thucydides in his introduction: "I have written my work, not as an essay which is to win the applause of the moment, but as a possession for all time." The new edition updates the coverage of the book to include topics that have become more central in the field in the more than two decades since the original edition was published. He has added several sections and many new applications and examples, and in doing so has made a great book even better.

Now that I have hopefully convinced you of the quality of the book, I should say a few words about what it contains. So what is Enumerative Combinatorics? Conveniently, that is the title of the first chapter and Stanley tries to give the simplest answer possible: "The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set… Immediately, philosophical difficulties arise. What does it mean to 'count' the number of elements of $S$? There is no definitive answer to this question." Stanley goes on to discuss the differences between explicit formulas, asymptotic formulas, generating functions, and other ways one might try to count finite sets. He then dives into examples, and very quickly the reader is taken on a tour of sets, multisets, cycles, partitions, Euler numbers, and permutations, all leading up to a detailed description of "The Twelvefold Way" (normally attributed to Rota) describing the various ways of counting functions between two sets.

The second chapter of the book deals with Sieve Methods, which Stanley describes as "a method for determining the cardinality of a set $S$ that begins with a larger set and somehow subtracts off or cancels out unwanted elements." His first example of such a method is the well-known Principle of Inclusion-Exclusion, which Stanley then uses to count sets such as derangements (permutations with no fixed points) and the number of ways to place nonattacking rooks on a chessboard. This latter topic leads to more generalisations such as Ferrers boards, Young diagrams, V-partitions, and involutions. A final section discusses how determinants of matrices have nice combinatorial interpretations.

Partially Ordered Sets (aka posets) are the topic of the third chapter, and Stanley discusses them in a variety of forms, including lattices, incidence algebras, and hyperplane arrangements. He then defines and studies the notion of $(P, \omega)$ partitions, which are partitions of an integer with certain restrictions on the parts given by inequalities. Other topics in this chapter include Eulerian posets, Binomial Posets, and Differential Posets. The fourth and final chapter of Stanley's book deals with counting problems whose answer can be given in terms of a rational generating function. In particular, Stanley gives a classification for when a generating function will actually be given by a polynomial or a quasipolynomial. He also discusses counting problems arising from counting solutions to linear Diophantine equations and gives applications such as counting the number of magic squares. Finally, he discusses the so-called Transfer-Matrix method which has many applications, including some in statistical mechanics.

Enumerative Combinatorics is a book that many people have picked up in order to learn something about the field, and it is excellent for that purpose. Stanley is an extremely clear writer, and there are more than enough examples and applications to give a real feel for what the theory is saying. Additionally, the book has 537 exercises, a number of which have ten or more parts, so there is no shortage of practice for a reader who wants to make sure they understand the theory well. Many of the exercises have hints or solutions, and they are all given a difficulty rating. Stanley has provided a very detailed bibliography and notes to point the interested reader in the proper direction to learn even more.

This is a graduate level textbook, and as such Stanley assumes a pretty high level of mathematical sophistication from his readers. However, the actual content that he assumes is fairly minimal and I suspect that a good undergraduate would also enjoy - and be able to learn quite a bit from - this book. As would a graduate student trying to learn combinatorics. Or an algebraic geometer who has stumbled across a combinatorial problem in their research. Or an expert in the field wanting to brush up on their background. Or just about any mathematician wanting to learn something new. And while I haven't set up the explicit mapping, I imagine that you fall into one or more of these sets.

8.3. Review by: David Jackson.
SIAM Review 55 (1) (2013), 193-194.

Within the twenty-five years that have passed since the publication of the first edition of Professor Stanley's Volume I of Enumerative Combinatorics, many advances have been made in this area. During this period there has developed a more general recognition of the presence of discrete, or combinatorial, structure at the heart of deep questions in many parts of mathematics and the mathematical sciences. The first edition of the book has clearly had an effect, both in engaging the interest of other mathematicians in the field and in setting out a productive view of part of the area. Like the first edition, the second edition is suitable for advanced undergraduates and beginning graduate students.

The opening of Chapter 1 describes the "Basic Problem" of enumerative combinatorics as

... that of counting the number of elements of a finite set. Usually we are given an infinite collection of finite sets $S_{i}$ where $i$ ranges over some index set $I$ (such as the set of non negative integers $\mathbb{N}$), and we wish to count the number $f(i)$ of elements in each $S_{i}$ "simultaneously." Immediate philosophical difficulties arise. What does it mean to "count" the number of elements of $S_{i}$? There is no definitive answer to this question. Only through experience does one develop an idea of what is meant by a "determination" of a counting function $f(i)$.

Examples of standard ways of describing the counting function $f(i)$ are presented: (a) an explicit closed formula for $f(i)$ involving only well-known functions and free of summation symbols; (b) a recurrence for $f(i)$; (c) an algorithm for computing $f(i)$; (d) an asymptotic estimate for $f(i)$; (e) a generating function for $f(i)$. The Basic Problem is clearly an exceedingly general one, and while specific problems of this type appear at first sight to be sui generis and illaqueating, their study over time has revealed underlying principles of great generality. The origins of many of the specific questions lie in other parts of mathematics and the mathematical sciences. Volume 1 narrows the enumerative material to manageable proportions so that some of the general principles are made apparent.

Many instances of the Basic Problem are found in other areas of mathematics and its applications, as such statistical mechanics, quantum field theory, algebraic geometry, and the study of knot invariants. The problem of analysing the running time of algorithms on a computer may be expressed precisely as an enumerative problem. The difficulty that presents itself is how to recognise large classes of enumerative problems, and how to provide a cogent and coherent account of general methods in a formalism that may be applied to these classes.

An instance of this is the formalism associated with partially ordered set that is the subject of Chapter 3. The "Principle of Inclusion and Exclusion" is widely known and may be regarded as a way of obtaining the solution to a particular question by solving a related question with fewer conditions. The two questions need to be related in a particular way, clearly, for this to be feasible. It is the theory of partially ordered sets that places the principle in a rich context that then provides the means of addressing questions of great complexity. A second example, from statistical physics, is the de termination of the binding energy of glue (a polymer, and therefore with long stringy chemical diagrams) between two narrowly separated parallel plates. The question may be modelled in terms of paths on an integer lattice, with unit steps weighted by physical constants and interaction terms, and then solved through the enumerative theory of paths, thereby avoiding the use of transfer matrices and the heavy computation they require.

There are two important concomitants of the general theory. The first is the notion of a combinatorial construction. This is a construction that expresses each element of $S_{i}$ bijectively in a simpler form while preserving designated enumerative information. An example of this is seen in the classical area of additive partition theory, where an integer partition is re-expressed in terms of the Ferrers diagrams obtained by deleting its maximal square. This part of enumerative combinatorics is often referred to as bijective combinatorics. The constructions range from the self-evident to the most delicate and subtle. The second important concomitant is the use of algebras or algebraic structures to preserve enumerative information and thence to obtain a generating series. An example is the use of the double coset algebra of the hyperoctahedral group in determining the generating series for two-cell embeddings of rooted graph in surfaces. In the earlier stages of the study of enumerative combinatorics, it was the ring of formal power series that played this role, without the use of an intermediating algebra.

The second edition is a considerable expansion of the first edition in several ways. The original chapter headings re main: What is Combinatorics?", "Sieve Methods," "Partially Ordered Sets," and "Rational Generating Functions." How ever, the scope of each chapter is extended. There is now a fuller account of additive partition theory from the point of view of enumerative combinatorics. The account of partially ordered sets, the topic of Chapter 3, has been considerably extended and now contains an account of differential posets and a section on transfer matrix techniques. A large number of exercises have been added in the second edition. These are invaluable for a true understanding of the material, and it is essential for a reader to work through some (many) of them to obtain a sense of how much the material presented in this volume can accomplish. In particular, it is in the exercises that additional bijective and algebraic material is adduced. Some easier exercises have been added by popular demand, and are certainly a useful addition. Solutions, and hints towards solutions, are provided for the harder questions.

The solutions also play another role that is easily overlooked at the introductory level, namely, a careful examination of the form of a generating series, and its constituents, and transformations that may be applied to it, may in turn reveal further insights into combinatorial structure itself. An in stance of this is the connexion between the generating series for rooted triangulations in orientable surfaces and the KP (Kadomtsev-Petviashvili) hierarchy. Deliberately sought outcomes of this sort from enumerative arguments are often overlooked by those whose attention is limited solely to the outcomes (a) to (e) listed above.

The notes following each chapter are very useful and have been extended. They pro vide not only historical information, but also brief accounts of alternative approaches to the theory and directions taken towards the more advanced material. Bibliographic references are included that enable the reader to pursue these topics further.

It is reasonable to assert that the second edition provides the reader with clear expositions of the main combinatorial structures that arise in enumerative combinatorics, bearing in mind that the second volume contains more advanced material.

Many decades ago, I was asked over lunch, in serious tones, why would one enumerate. Caught athwarts, I looked for subtext of the question, suppressed the classicist's Cur? - Cur non? formulaic response, and resorted to a limp suggestion that it would take too long to explain adequately in the circumstances. On reading this review, I wish now that I would have had it then, in my pocket, to produce with a flourish. In short, Stanley's second edition is an exceedingly welcome addition to the canon of enumerative combinatorics.

9.1. From the Publisher.

Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. The combination of the author's extensive knowledge of combinatorics and classical and practical tools from algebra will inspire motivated students to delve deeply into the fascinating interplay between algebra and combinatorics. Readers will be able to apply their newfound knowledge to mathematical, engineering, and business models.

The text is primarily intended for use in a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory. Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and rudiments of group theory. The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix-Tree Theorem, de Bruijn sequences, the Erdős-Moser conjecture, electrical networks, and the Sperner property. There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labelled trees.

9.2. Review by: Felipe Zaldivar.
Mathematical Association of America (14 December 2015).
https://maa.org/press/maa-reviews/algebraic-combinatorics-walks-trees-tableaux-and-more

There are plenty of undergraduate books on graph theory and discrete mathematics in general. There are fewer books with an algebraic approach. If one adds as extra requisites that the book is well-written or elegant in its choice of topics, the field becomes even more rarefied.

The book under review is one of those few exceptions. The chosen topics represent a sample of enumerative combinatorics suitable for the elementary algebra available to an undergraduate student. At the same time, this selection highlights the power of the algebraic method to obtain deep and interesting combinatorial results. The motivated student will be well prepared and willing to learn more algebra, even when her or his motivation lies in the combinatorial realm.

Perhaps a few samples will illustrate the book approach. In Chapter 1, page 1, we find the definition of a graph, its adjacency matrix, and loops in a graph. The first remark, on top of page 2 is that this adjacency matrix is real symmetric and its trace is the number of loops in the given graph. A nicely chosen example illustrates this remark. Next, we find the definition of a walk of length $<curlyl>$ in a graph and immediately after the theorem that the $(i, j)$-entry in the $ℓ$-th power of the adjacency matrix of a given graph is the number of walks of length $ℓ$ from the $v_{i}$ vertex to the vertex $v_{j}$. The proof is just an interpretation of the formula for an entry in a product of matrices. The rest of this first chapter explores some consequences of the above theorem, essentially depending on the eigenvalues of the adjacency matrix.

A second sample is also classical: Pólya's theory is introduced in Chapter 7. Here the problem is the enumeration of objects when the set of such objects is subject to the action of a finite group. Again, after some classical motivation using colourings of the combinatorial objects, Burnside's lemma is proven and from it several consequences are obtained, including of course, Pólya's theorem and one of its better known corollaries that count the number of coloured necklaces of given length.

I could go on listing the combinatorial jewels in each chapter of this book, but let me just finish by saying that indeed, all the combinatorial topics have been chosen for the linear algebra and elementary group and field theory to shine and advertise their power. This is a book that can be used to teach a topics course for senior undergraduates. It will help them to solidify their just-acquired abstract algebra and at the same it will introduce them to some beautiful topics in pure or applied combinatorics.

9.3. Review by: D V Feldman.
Choice: Current Reviews for Academic Libraries 51 (8) (2014), 1442.

An undergraduate education in pure mathematics progresses toward, but should not stop at, the three pillars of abstract algebra, (real and complex) analysis, and (general and geometric) topology. These subjects mostly equip the student with tools, but tools only prove their mettle when they solve problems that precede them (conceptually, at least). Many questions of enumerative combinatorics do have direct intuitive appeal, and this book puts basic abstract algebra to work at cracking them. Stanley is a renowned expert on the subject; his Enumerative Combinatorics stands as the bible for graduate students. While many topics treated in those volumes could fall within the ken of an undergraduate, most would find the pace and density daunting. This gentle book provides the perfect stepping-stone up. The various chapters treat diverse topics, e.g., random walks, chains and anti-chains in lattices, partitions, Pólya counting, the Robinson-Schensted-Knuth algorithm, Eulerian tours, and electrical networks relating to geometrical decompositions. Stanley's emphasis on "gems" unites all this - he chooses his material to excite students and draw them into further study. As prerequisites, linear algebra and the basics of finite groups should suffice. Summing Up: Highly recommended. Upper-division undergraduates and above.

10.1. From the Publisher.

Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their properties and applications to combinatorics, algebra, analysis, number theory, probability theory, geometry, topology, and other areas. Following an introduction to the basic properties of Catalan numbers, the book presents 214 different kinds of objects counted by them in the form of exercises with solutions. The reader can try solving the exercises or simply browse through them. Some 68 additional exercises with prescribed difficulty levels present various properties of Catalan numbers and related numbers, such as Fuss-Catalan numbers, Motzkin numbers, Schröder numbers, Narayana numbers, super Catalan numbers, $q$-Catalan numbers and $(q, t)$-Catalan numbers. The book ends with a history of Catalan numbers by Igor Pak and a glossary of key terms. Whether your interest in mathematics is recreation or research, you will find plenty of fascinating and stimulating facts here.

10.2. From the Preface.

This text had its origins in the 1970s, when I first started teaching enumerative combinatorics and became aware of the ubiquity of Catalan numbers. Originally I just made a handwritten list for my own benefit. One of the earliest such lists has survived the ravages of time and appears in Appendix A. Over the years, the list became larger and more sophisticated. When I wrote the second volume of Enumerative Combinatorics (published in 1999), I included sixty-six combinatorial interpretations of Catalan numbers (Exercise 6.19) as well as numerous other exercises related to Catalan numbers. Since then I have continued to collect information on Catalan numbers, posting most of it on my "Catalan addendum" web page. Now the time has come to wrap up this 40+ years of compiling Catalan material, hence the present monograph. Much of it should be accessible to mathematically talented undergraduates or even high school students, while some parts will be of interest primarily to research mathematicians.

This monograph centres on 214 combinatorial interpretations of Catalan numbers (Chapters 2 and 3). Naturally some subjectivity is involved in deciding what should count as a new interpretation. It would be easy to expand the list by several hundred more entries by a little tweaking of the current items or by "transferring bijections." For instance, there is a simple bijection $\phi$ between plane trees and ballot sequences. Thus, whenever we have a description of a Catalan object in terms of plane trees, we can apply $\phi$ and obtain a description in terms of ballot sequences. I have used my own personal tastes in deciding which such descriptions are worthwhile to include. If the reader feels that 214 is too low a number, then he or she can take solace in the solution to item 65, which discusses infinitely many combinatorial interpretations.

Also central to this monograph are the sixty-eight additional problems related to Catalan numbers in Chapters 4 and 5. Some of these problems deal with generalisations, refinements, and variants of Catalan numbers, namely, $q$-Catalan numbers, $(q, t)$-Catalan numbers, $(a, b)$-Catalan numbers, Fuss-Catalan numbers, super Catalan numbers, Narayana numbers, Motzkin numbers, and Schröder numbers. Here we have made no attempt to be as comprehensive as in Chapters 2 and 3, but we hope we have included enough information to convey the flavour of these objects.

In order to make this monograph more self-contained, we have included a chapter on basic properties of Catalan numbers (Chapter I) and a Glossary. The Glossary defines many terms that in the text are simply given as citations to the two volumes of Enumerative Combinatorics. ...

The history of Catalan numbers and their ilk is quite interesting, and I am grateful to Igor Pak for contributing Appendix B on this subject. There the reader can find much fascinating information on a subject that has not hitherto received adequate attention.

10.3. Review by: David Callan.
Mathematical Reviews MR3467982.

This monograph will be the definitive reference on Catalan numbers for years to come. Its centrepiece is a listing of some 214 combinatorial manifestations of the Catalan numbers in problem and solution format, all presented in Stanley's inimitable style, with detailed references. There is supplementary material on the appearance of Catalan and related numbers in various contexts, some quite advanced, a brief polemical history of the Catalan numbers by Igor Pak, and a scan of the 1970s handwritten notes from which the book grew - a nice touch. The subject matter is still very active; for example, a fully bijective proof for item A.33(b) appeared only recently [P Hajnal and G V Nagy]. As usual with Stanley's books, the reader can jump in anywhere. The text is free of the misprints that were so oddly abundant in Enumerative combinatorics. Two minor quibbles: including the de Bruijn-Morselt bijection among the "simple" ones belies the tortuous history of this bijection; only one problem has a difficulty rating of 4. Overall, a delightful book.

10.4. Review by: Gerry Leversha.
The Mathematical Gazette 101 (551) (2017), 377-378.

The Catalan numbers are probably the most ubiquitous sequence of numbers in combinatorial mathematics, since they show an uncanny propensity to crop up in many different contexts. They describe such diverse situations as handshaking patterns around a table, binary trees, mountain profiles, ballot sequences, Dyck paths, parenthesisations in non-associative algebra and triangulations of polygons. The author of this short monograph confesses to an addiction to this topic since he started teaching enumerative combinatorics in the 1970s, and he included 66 interpretations of Catalan numbers in his 1999 book published on that subject. His new offering increases this number to 214, although, as he readily admits, some subjectivity is involved in deciding what should count as different examples.

The fundamental idea is that of bijection. Once you know that ballot sequences, for example, yield the Catalan numbers, the fact that you can create a one-to-one correspondence between these and planar trees of a particular type is enough to show that the sequence also occurs in this context. On the other hand, it is tempting to prove this directly rather than by appealing to bijection, and an alternative approach is to show that different scenarios yield the same recurrence relation.
...
This is a novel and intriguing little book which will appeal to anyone who appreciates problem-solving and is fascinated by the connections between different mathematical ideas.

11.1. From the Publisher.

Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. The combination of the author's extensive knowledge of combinatorics and classical and practical tools from algebra will inspire motivated students to delve deeply into the fascinating interplay between algebra and combinatorics. Readers will be able to apply their newfound understanding to mathematical, engineering, and business models. Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and rudiments of group theory. The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix-Tree Theorem, de Bruijn sequences, the Erdös-Moser conjecture, electrical networks, the Sperner property, shellability of simplicial complexes and face rings. There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labelled trees.

The new edition contains a bit more content than intended for a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory. Instructors may pick and choose chapters/sections for course inclusion and students can immerse themselves in exploring additional gems once the course has ended. A chapter on combinatorial commutative algebra (Chapter 12) is the heart of added material in this new edition. The author gives substantial application without requisites needed for algebraic topology and homological algebra. A sprinkling of additional exercises and a new section (13.8) involving commutative algebra, have been added.

12.1. From the Publisher.

This book features mathematical problems and results that would be of interest to all mathematicians, but especially undergraduates (and even high school students) who participate in mathematical competitions such as the International Math Olympiads and Putnam Competition. The format is a dialogue between a professor and eight students in a summer problem solving camp and allows for a conversational approach to the problems as well as some mathematical humour and a few nonmathematical digressions.

The problems have been selected for their entertainment value, elegance, trickiness, and unexpectedness, and have a wide range of difficulty, from trivial to horrendous. They range over a wide variety of topics including combinatorics, algebra, probability, geometry, and set theory. Most of the problems have not appeared before in a problem or expository format. A Notes section at the end of the book gives historical information and references.

12.2. From the Preface.

The original inspiration for this book was a series of bridge (the card game) books by David Bird. These books feature a mixture of amusing characters (such as a pompous abbot, a clever parrot, and Robin Hood) and interesting bridge hands. Could something similar be done for mathematics? If so, what should be the subject matter? My answer to this question is the present book.

For many years at M.I.T. I taught or co-taught the freshman seminar 18.S34, later called 18.A34, on Mathematical Problem Solving, focused on the Putnam Competition. During the course of teaching this seminar I accumulated lots of interesting problems and mathematical facts, as well as some dumb jokes. This provided most of the material herein.

The intended audience for this book is mathematicians at all levels beginning with undergraduates, and even high school students, who are adept at solving challenging problems such as those appearing on the IMO or Putnam Competition. The primary purpose of the problems in this book is not didactic, but rather to entertain. Nevertheless I hope that at least some readers will learn some interesting and useful mathematics. For readers primarily interested in the mathematical content of this book, I have included a List of Problems preceding each chapter.

12.3. From the First Day.

Professor Mortimer Ignatius Blakeley walked into Room C-122 on the first day of Problem Solving Camp. Eight eager but somewhat apprehensive aspiring mathematicians were seated at their desks. Professor Blakeley already knew quite a bit about these students from having read their files during the selection process and from the reception the previous night. They had been chosen from a highly talented group of undergraduate math majors throughout the USA, and now the time had finally come to see how they would do!
...
"Welcome to Problem Solving Camp!" said Professor Blakeley. "I am sure you will find this a great opportunity to learn new mathematics and to enjoy working on some challenging problems. You are already familiar with the course mechanics. To quickly review, in the mornings we will discuss mathematics and work on some new problems. In the afternoon session I'll pass out some problems related to the morning's discussion, and we can work on them with the help of a couple of course assistants. We can also discuss progress on earlier problems that have not yet been solved. Now let us get down to serious business!"

12.3. Review by: Edward J Barbeau.
Mathematical Reviews MR4249565.

Eight bright and lively undergraduate students led by an erudite professor in a problem-solving seminar engage in sixteen imaginary conversations that range over a variety of mathematical problems - some recreational, some surprising and some deep. Each session begins with a problem that sparks a discussion wending through possible solutions, related questions, pertinent results and mathematical lore, salted with a bit of humour. The content is appropriate for undergraduates with a serious interest in mathematics and a solid background in the core of the undergraduate syllabus: linear algebra, polynomials, analysis, probability, foundations and geometry. Much of the material is slanted towards combinatorics, with attention paid to enumeration, sequences, recursions, generating functions and tilings.

Each chapter is packed with a substantial amount of mathematics, both traditional and contemporary, that cannot be rushed through. Some results are proved in the course of the conversation, while references are provided for others. There is much to delight in and to savour in the fecundity of material. This book, building on the author's own experience running a freshman seminar on problem solving, will suggest possibilities for other supervisors of such activity.

12.4. Review by: D V Feldman.
Choice: Current Reviews for Academic Libraries 58 (12) (2021), 1206.

Stanley posits a dialogue in which a fictional, slightly pompous professor challenges his eight fictional, genius-level, and diverse students and rarely stumps them all. The amusing conversations that ensue teach valuable problem-solving tricks that mathematicians use constantly, yet rarely try to systematise. Although the answers here generally arrive immediately, they come tersely, sometimes more like hints, so that a careful reading - not so passive after all - will actually demand mental effort much like digesting any theorem/proof book. A few old saws recur in the text, but they serve as points of departure for genuine novelties.