Suggested Reading for Combinatorics

manpreet Best Answer 2 years ago

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