How to Count: An Introduction to Combinatorics, Second Edition (Discrete Mathematics and Its Applications) 2nd Edition by R.B.J.T. Allenby (PDF)

Description

Emphasizes a Problem Solving ApproachA first course in combinatoricsCompletely revised, How to Count: An Introduction to Combinatorics, Second Edition shows how to solve numerous classic and other interesting combinatorial problems. The authors take an easily accessible approach that introduces problems before leading into the theory involved. Although the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics.New to the Second EditionThis second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet’s pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises. Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Pólya’s counting theorem.

User’s Reviews

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

⭐Great reference book for any body working on combinatorial optimization

⭐Let me start off with what I like about this book.The topic coverage is great and the order of presentation of material seems sensible. The printing is clear although the positioning of diagrams at the top or bottom of pages is awkward. In the foreword the authors recognise this layout issue and apologise for this being out of their control (see below). I like their approach to providing proofs to the theorems.I really like the clear and early statement of “The Principle of Multiplication of Choices”. This makes it clear how to combine simple choices into a more complex situation. This is very widely applicable in combinatorial problems. However, it is not made clear if this principle is a theorem that can and needs to be proved or is an axiom that just needs to be accepted and used.I get the impression the authors are expert in this field and have made a great effort to organize and explain the material. It doesn’t work for me.A big stumbling block for me occurs early on after Problem 2.4 where the idea that to go from a permutation to a combination one divides by a certain number. Although this seems plausible in the examples given, in my mind it isn’t made clear as to why this is valid. Is this a theorem or an axiom?The positioning of diagrams not being next to the text that refers to the diagram is maddening. The argument is it down to the computer program is bogus. Computer type and page setting gives you as much control over layout as manual systems. The reason using the top or bottom of page for the location of diagrams is this is an easy/lazy way to avoid white space on the page before or after diagrams. However, if you look at Tufte’s book you can see what can be achieved with layout. In fact Tufte comments he rejected conventional publishing because of this unnecessary and ugly imposition on page layout.Although the authors have tried to provide combinatorial style proofs there is no introductory guidance as to what comprises a combinatorial proof nor how they are arrived at. The proofs are very wordy with no supporting diagrams therefore they are extremely difficult to follow and understand. However, in section 6.2 they relent and provide diagrams that support the proofs. Yes, being rigorous the diagram cannot be the proof but a diagram can clearly provide the motivation for the proof and guide the reader in understanding the proof. Next, a small point, but when you read the proofs there is no indication the proof has ended and you are back in the text. The convention is to use the black square (Halmos) to indicate the end of the proof and maybe leave an extra half line spacing after the end of the proof. Neither are provided and this is another irritation when reading the proofs. Most of the proofs in the book are written in the order: motivation, statement of theorem, proof. However, the first two theorems in the book, 2.1 and 2.2 , don’t follow this pattern and it took some searching to find the proof precedes the theorem in these two cases. Strange.I find the style of writing a little archaic and jarring. For example at the beginning of section 10.3 “In this section we discuss some practical problems to which graph theory contributes solutions”. Why “contributes” this suggests something incomplete, why not “provides”? Also words are used with which the authors assume the reader is familiar. For example, “hand of cards” in problem 2.5. It isn’t made completely clear that this is a selection of cards where order is not important, a feature that is essential to the solution of the problem. Yes, ” … the order in which the cards is chosen does not affect the hand we end up with” is there but I feel it would have been better to define a hand of cards “.. a selection of cards from a complete set (or pack) of playing cards, where the order is not relevant”. Then hands of cards would have been firmly rooted in the “combination” camp. A final example is on page 19, “The above remarks yield the general formula for P(n,r)”. Why not, “The argument above justifies the general formula for P(n,r)”? Of course, my comments about wording may be dismissed as just a matter of personal taste.Finally, after the proof theorem 2.3 the authors start discussing sub-sets. But the link between selections and sub-sets is not made clear and I am left wondering why and how these sub-sets relate to the situation being discussed.So I think the book contains most of what you would want to know about combinatorics, I just feel it could have been presented and written better.

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