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I am an undergraduate wiht basic knowledge of combinatorics, but I want to obtain sound knowledge of this topic. Where can I find good resources/questions to practice on this topic?

I need more than basic things like the direct question 'choosing r balls among n' etc.; I need questions that make you think and challenge you a bit.

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    $\begingroup$ Richard Stanley's Enumerative Combinatorics and Wilf's Generatingfunctionology are both excellent books by excellent authors, you can't go wrong. Both are available freely online from the authors' web pages. Disclaimer: I have never seriously studied out of them, I've used them more as references when I need to learn about a particular topic, but my experience is that they are excellent. $\endgroup$ – Ragib Zaman Jul 16 '12 at 15:07
  • $\begingroup$ I couldn't find links of book on author's webpage . Could u please provide me link. $\endgroup$ – user35921 Jul 16 '12 at 17:04
  • $\begingroup$ @tesla For the second, look at my answer below. $\endgroup$ – M Turgeon Jul 16 '12 at 17:26
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    $\begingroup$ Here is Generatingfunctionology and here is Volume 1 of Enumerative Combinatorics. Wilf's book is at a more elementary level than Stanley's, with plenty of do-able exercises you can practice one, so is better for you to learn from. Stanley is still a great read and reference, though is aimed more towards graduate level students, and many of the exercises are extremely difficult problems (most of the solutions are in journals). $\endgroup$ – Ragib Zaman Jul 17 '12 at 3:11
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    $\begingroup$ Martin Aigner's book "A Course in Enumeration" is quite accessible and contains many good exercises. I would recommend you check it out if your library has a copy. $\endgroup$ – Daniel Pietrobon Aug 19 '12 at 6:46
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As far as book are concerned, my favorite basic combinatorics books are Basic Techniques of Combinatorial Theory by Daniel I.A. Cohen and Combinatorics and Graph Theory by Harris, Hirst and Mossinghof. Cohen, in particular, is a great resource for questions which will make you think deeply and expand your horizons. Most chapters have well over 70 exercises, ranging from rountine to quite difficult. Since the quality of questions seems paramount to your decision I will include an example exercise from Cohen:

A collection of $n$ lines in the plane are are said to be in general position if no two are parallel and no three are concurrent. Let $a_n$ be the number of regions into which $n$ lines in general position divide the plane. How big is $a_n$?

And this is just the tip of the iceberg. Cohen's book is full of high-quality exercises, most of which have attributions to the originators.

Something that really sets Cohen's treatment of the topic apart from others is the fact that he often gives 2 or 3 different proofs of theorems. Unfortunately, the book is out of print; however, there are still many used copies for sale on Amazon.

Combinatorics and Graph Theory by Harris, Hirst and Mossinghof covers much of the same basic combinatorial material as Cohen. To me, what really sets this book apart is the inclusion of infinitary combinatorics, particularly their treatment of Ramsey theory.

If you are looking for a more advance treatment of combinatorics, then you will find Enumerative Combinatorics by Richard Stanley more than accommodating, with hundreds of difficult exercises, some of which (at least at the time of writing) are unsolved.

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  • $\begingroup$ +1 for Combinatorics and Graph Theory by Harris. It is a very good book for a first course. $\endgroup$ – Shahab Jul 16 '12 at 14:32
  • $\begingroup$ @Shabab Totally agree and +1 for the recommendation of this fine book. $\endgroup$ – Mathemagician1234 Jul 16 '12 at 22:33
  • $\begingroup$ Is there a free copy of this book available ? $\endgroup$ – user35921 Jul 17 '12 at 5:49
  • $\begingroup$ @tesla If you give me an email address I can email you a pdf copy. $\endgroup$ – Holdsworth88 Jul 17 '12 at 5:50
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    $\begingroup$ @Holdsworth88 could you please mail me a pdf copy of Basic Techniques of Combinatorial Theory by Daniel I.A. Cohen. maybe you just upload it on dropbox and give a link. Thanks. $\endgroup$ – user173094 Dec 18 '14 at 0:57
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A great book focused mainly on generating functions is generatingfunctionology by H. Wilf. Also, have a look at Peter Cameron's webpage; he has several lecture notes and a link to his book Combinatorics - Topics, Techniques, Algorithms.

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The text "A Walk through Combinatorics" by Miklos Bona is a well-written and understandable introduction, in both my opinion and the opinion of the couple of people who have used it. The book covers a wide range of topics so you can get a taste of many different parts of combinatorics. It has some "core" material basic to enumerative combinatorics as well as basic graph theory, but also has a "Horizons" section with treatments of more advanced (or at least less central) topics like ramsey theory, pattern avoidance in permutations, and algorithmics/computational complexity.

I've also heard that while Stanley's "Enumerative Combinatorics" is a comprehensive, standard book, it is somewhat terse and may not be the easiest to work through on your own.

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  • $\begingroup$ +1 for Bona-one of 2 excellent textbooks by him on combinatorics and graph theory from a modern point of view. $\endgroup$ – Mathemagician1234 Aug 19 '12 at 7:40
  • $\begingroup$ I'm currently using it for self-study, and it is indeed a fun and accessible introduction to combinatorics. There is an errata on Bona's website, but there are still a number of critical errors that haven't been caught though. (I'm referring to the older, 2nd edition of the text.) $\endgroup$ – Ryan G Jul 11 '13 at 14:57
  • $\begingroup$ @Ryan: Alas Aigner's book which has been offered as an alternative is also not free of critical errors; there's one on p. 25 for example. And I'm not sure it even has an errata list somewhere. The main issue with Bona's book is the rather confusing presentation at times... $\endgroup$ – Fizz Apr 4 '15 at 13:48
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Lots of good suggestions here. Another freely available source is Combinatorics Through Guided Discovery. It starts out very elementary, but also contains some interesting problems. And the book is laid out as almost entirely problem-based, so it useful for self study.

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Apart from the book suggestions given here, you may also like to take a look at MIT OCW's Combinatorics: The Fine Art of Counting.

Although designed for High school students, few problems that might make you think.

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A good book is Discrete mathematics by N. Biggs. His writing style is very clear, it offers many good exercises (with solutions online) and covers interesting topics.

As for exercises I am in the process of creation of a page with exercises from various fields of mathematics. Check http://exwiki.org . The list is currently small, but you can post useful exercises that you will encounter along the way when studying combinatorics.

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Combinatorics underpins a branch of math called "finite mathematics" (the study of finite sets). I would recommend getting a book, or taking a course in finite mathematics.

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    $\begingroup$ @ Tom Au: a finite number of sets? I suppose you mean finite sets.. $\endgroup$ – boumol Jul 16 '12 at 15:23

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