What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a first-year-student in college. Do you know a good book on the subject?
Thanks.
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I had my first intro graph theory and combinatorics class last semester. The book we were using was pretty terrible so I looked around and found a copy of Combinatorics and Graph Theory by Harris et. al. and I really enjoyed it. The book contains a lot of topics and the explanations are very to the point. I especially liked the sections on Ramsey numbers.
The questions are all to the point and illustrate some important concept which is also nice. For example in the section on the happy ending problem the exercises reconstruct several historical proofs and introduce you to other problems like the empty polygon problem.
Here is a very positive review I read recently: http://www.maa.org/reviews/combinatoricsgraphs.html
Edit: I'm not sure if this book is appropriate for your situation specifically, but I highly recommend it none the less.
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My personal favorites are the following:
(is not always that introductory, but for those who like counting, it is a must have)
If you want really easy, but still interesting books, you might like Brualdi's book (though apparently, that book has many mistakes). Also interesting might be some chapters from Feller's book on Probability (volume $1$).
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Try Principles and Techniques in Combinatorics by Chen Chuan Chong and Koh Khee Meng or Combinatorics by Peter Cameron. The latter is more advanced and has more topics.
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A Path to Combinatorics for Undergraduates, by Titu Adreescu and Zuming Feng introduces the subject by presenting a large number of problems (many from Olympiads and other competitions), and covers a broad range of methods and results.
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There are some excellent combinatorics books which also look at the applicability of combinatorics both within mathematics and outside of mathematics:
a. Applied Combinatorics by Fred Roberts
b. Applied Combinatorics by Alan Tucker
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Something that often gets ignored is Schaum's Outlines: Combinatorics by V. K. Balakrishnan. Lots of small examples that are manageable by a beginner, shows you 'how to do it' in a straightforward manner.
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I'm quite fond of Benjamin and Quinn's Proofs that Really Count: The Art of Combinatorial Proof.
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This is not precisely what you want, but you could look at "Generatingfunctionology". The second edition is free, and can be downloaded here http://www.math.upenn.edu/~wilf/DownldGF.html
The book is about generating functions, which are helpful in combinatorial arguments.
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Not already mentionned (oldie but goodie):
A course in Combinatorics by van Lint and Wilson (book cover with card suits).
A lot of small chapters, some challenging concepts, basic graph, coding and design theory.
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Try also Notes on Introductory Combinatorics by Polya,Tarjan, and Woods. An earlier version is freely available online.
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Applied Combinatorics by Alan Tucker is a good one. It's short, not hard to follow, a lot of problems to work through, and it's split into two sections: graph theory in section 1, and combinatorics (generating functions, counting techniques, etc) in section 2.
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I’m fond of Miklós Bóna, Introduction to Enumerative Combinatorics; it’s extremely well written and doesn’t require a lot of background. Of the books that have already been mentioned, I like Graham, Knuth, & Patashnik, Concrete Mathematics, isn’t precisely a book on combinatorics, but it offers an excellent treatment of many combinatorial tools; it probably requires a little more mathematical maturity than the Bóna. A good next step beyond the relatively elementary level is Wilf, generatingfunctionology. Tucker, Applied Combinatorics, is very elementary but gives a decent taste of a very wide range of combinatorial topics.
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I'm surprised no one has mentioned S. Lando's "Lectures on Generating Functions". It's a great book that grew out of freshman seminars at the Independent University of Moscow. Granted, the freshmen at IUM are much more advanced than usual, but the book itself is written in a lucid manner and has great problems to think about. Literally a 10/10 choice for a first introduction to combinatorics.
If the student's leaning towards computer science at all, I'd recommend Knuth et. al.'s Concrete Mathematics. It's full of solid math and has the aim of building mathematical tools for CS. Other than that, a newer addition that looks promising is Russell Merris's "Combinatorics, 2nd ed." It gives a pretty broad introduction while also giving in-depth work and examples.