Introduction to Combinatorial Analysis (Dover Books on Mathematics) by John Riordan (PDF)

Description

This introduction to combinatorial analysis defines the subject as “the number of ways there are of doing some well-defined operation.” Chapter 1 surveys that part of the theory of permutations and combinations that finds a place in books on elementary algebra, which leads to the extended treatment of generation functions in Chapter 2, where an important result is the introduction of a set of multivariable polynomials.Chapter 3 contains an extended treatment of the principle of inclusion and exclusion which is indispensable to the enumeration of permutations with restricted position given in Chapters 7 and 8. Chapter 4 examines the enumeration of permutations in cyclic representation and Chapter 5 surveys the theory of distributions. Chapter 6 considers partitions, compositions, and the enumeration of trees and linear graphs.Each chapter includes a lengthy problem section, intended to develop the text and to aid the reader. These problems assume a certain amount of mathematical maturity. Equations, theorems, sections, examples, and problems are numbered consecutively in each chapter and are referred to by these numbers in other chapters.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a text which, while considered an introduction and perhaps a bit dated nowadays, requires some of the closest attention I have ever had to pay to a mathematical work. If I may call this book a mathematical gold mine, I must also say that in parts the digging is arduous and rewarding and the rock can be of high density. The section in chapter 2, on derivatives of composite functions (essentially Bell Polynomials, after the famous Eric Temple Bell), is an example.This is at least my third pass through the book and this time I decided not to slough over some of the details on these polynomials. I found myself bewildered for a couple of days, as the author gives no assistance to the reader–no doubt to help the reader develop some of that mathematical maturity mentioned in the preface–and found gold thereby. Statements made apparently as an aside, such as “completely determined…” had not been paid adequate attention by me. Also, it was not pointed out by the author that certain constants evaluated in the examples were found by setting the variable to zero, and so on. This is all left to the reader to dig out. In a span of two or three pages in this section there are hours of profitable work left to the reader. This is not a complaint, but rather a statement of the level of audience addressed by this book. It is not a text for readers who require hand-holding. However, anyone who is prepared to dig will experience the “joy of finding things out”, if I may humbly steal a favored phrase of the late, great Richard P. Feynman.

⭐With cheap computers and big databases, computer-intensive distribution-free statistical algorithms (non-parametric procedures, bootstrap methods, permutation approaches, etc.) are becoming much more common. Older methods mostly assumed one of several mathematically convenient underlying distributions (Normal, Poisson, Lognormal, and a few others) so they could be done with pencil and paper. Today we can use computationally intensive methods that avoid assuming a specific distribution.Many modern statistical procedures boil down to counting possibilities from very large but finite probability spaces. In a word, combinatorics. This book introduces the fundamental concepts behind these algorithms.

⭐The thing that borthers me is the archadic notation used in parts of this book and notation used with no reference that I can find.I really learned a lot in reading this book and intend to try to get more out of it in the future, but I really wouldn’t use this as an introductory text as some of the theorems and problems are graduate or post graduate level. It is a useful and helpful book for those intent on understanding combinatorial theory.

⭐I’ve got no basic knowledge on combinatorics,and I saw the word”introduction”,and therefore reckon it to be elementary and simple,tends deep gradually. It actually doesn’t meet my expectation. It’s like LandauLifschitz:go thru the materials so fast that u feel that the author thinks”why do u still not know it?” it claims itself to be an intro,but really not introductory. I don’t know if it’s good as a review book, but I believe it’s not a good first book.

⭐I was looking for some very specialized stuff, and this book had been cited as the source. Sure enough, there it was, and that turned out be a big help. So I got what I needed, which is way I had ordered it.L. Drooks

⭐It could have used more practical examples. Otherwise, a great book!

⭐This is an old (1958) text on what we can call advanced combinatorics. The book can be viwed as a transition text between the old tradition in combinatorics and the then new methods that were becoming fashionable. Notation is slightly “peculiar” for modern standards.While the text contains derivations of many valuable results, the style is terse and the author definitely does not strive to be clear. It may take a day to advance a couple (I really mean a couple: 2) of pages of text if you really intend to absorb the material. Some exercises are hard, all without solutions. In my humble opinion, the value of the book lies in its chapters 4, 7, and 8 on permutations with restrictions that the author explores in more detail than in most other presentations available, although, again, in a terse manner. Very little is said about the symmetric group and symmetric functions. The author has also published another book on combinatorics that is equally useful and terse: J. Riordan, “Combinatorial Identities” (New York: Krieger, 1979).Maybe it would be a good idea to “rewrite” the book in modern notation, keeping the order of presentation and choice of subjects but expanding the demonstrations, clarifying the arguments, and selecting and solving the exercises. The textbook by C. L. Liu, “Introduction to Combinatorial Mathematics” (McGraw-Hill, 1968) closely follows the presentation of Riordan but gives more examples. Another option is the text by C. A. Charalambides.If you do not intend to specialize in combinatorics, I’d recommend a more modern treatment like the ones in the books by J. M. Harris, J. L. Hirst, and M. J. Mossinghoff, ”

⭐,” for beginners, and J. H. van Lint and R. M. Wilson, ”

⭐,” if you are more seasoned. If you want to do research, look elsewhere (Stanley, Lovasz, etc.). The text by Riordan, however, is still valuable if you stumble on some problem involving restricted permutations, permanents, and rooks (and related problems involving decks of cards…).

⭐A very good guide to the classical theory of combinatorics.As it dates from 1958 there is no mention of computer algorithms or software,but is excellent as a theoretical introduction.

⭐A classic . Worth every penny

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