What are some suggestions for texts on introductory combinatorics and its applications? I would prefer if the applications would be to other branches of mathematics rather than outside of mathematics.

I've been interested in math for awhile and am currently in my third year of studying it at a university but strangely, I've managed to avoid the subject in the process and given my school's limited curriculum, there is no combinatorics course.

Frankly, I feel rather illiterate with out knowing something of the subject, especially seeing it in other areas; on the other hand, I am not completely fascinated by the subject so I'm hoping for something concise and straightforward with "suitable" amount of detail.

If it could cover some basic and fundamental theorems, some problem solving strategies using combinatorics, as well as some introductory topics in graph theory, that would be great.

Sorry for giving rather subjective criterion; any suggestions are welcome.

Thank you in advance!

Edit: I took a look at the syllabus of an introductory combinatorics course taught in Hungary. I figured that since the Hungarians are rather strong in this subject, maybe I should go with what they deem as "introductory." I don't know if that would be too optimistic of me. Here is a list of the topics. Most of it is unfamiliar to me.

-Basic counting rules (product rule, sum rule, permutations, combinations, Pascal's triangle, occupancy problems, distribution problems, Stirling numbers).

-Generating functions (definition, operations on generating functions, applications to counting, binomial theorem, exponential generating functions).

-Recurrences (Fibonacci numbers, derangements, the method of generating functions).

-Principle of inclusion and exclusion (the principle and applications, occupancy problems with distinguishable balls and cells, derangements).

-Graph theory taster (overview of fundamental concepts, connectedness, graph coloring, trees, Cayley's Theorem on the number of trees, Eulerian circuits and their applications: de Bruijn cycles, planar graphs).

-Pigeonhole principle and Ramsey theory (Ramsey's theorem, bounds on Ramsey numbers, applications).

-Symmetric combinatorial structures, block designs (definition, Latin squares, finite projective planes).

-Two fundamental theorems on set systems: Erdös-Ko-Rado Theorem and Sperner's Theorem

  • $\begingroup$ How introductory that should be? There is an excellent set of notes by Jacob Lurie, you may find it useful: link. $\endgroup$ – Artem Nov 12 '14 at 2:32
  • $\begingroup$ @Artem: That's a good question. I was looking at the syllabus of an introductory course in combinatorics in Hungary. Since the Hungarians seem rather strong in this subject and since they deem that particular course as introductory, I'll go with what they include in the syllabus. I'll make an edit to the question. $\endgroup$ – inkievoyd Nov 12 '14 at 16:38
  • $\begingroup$ Almost everything, if I recall correctly, is in Lurie's notes. They will require some work, but give a very solid background. $\endgroup$ – Artem Nov 14 '14 at 15:55

I think a text like Applied Combinatorics by Alan Tucker will not only be really applicable (and useful), but will really spark your interest with the subject. The text isn't too proof-heavy.

If you use that with Benjamin & Quinn's Proofs that Really Count, and excellent book on combinatorial proofs, I think you'll gather a taste and introductory-to-intermediate grasp on the topic.

Combinatorics is really hard not to fall in love with, as the topic contains some of the most "beautiful" proofs mathematicians have come up with.

  • $\begingroup$ Perhaps I shall become completely fascinated by the subject then! Thank you for the recommendations. $\endgroup$ – inkievoyd Nov 11 '14 at 18:09
  • $\begingroup$ @inkievoyd you're welcome $\endgroup$ – daOnlyBG Nov 11 '14 at 18:25

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