I'm curious about combinatorial analysis. I have some questions about this topic because when I think of combinatorics I recall some technics to solve problems but that's all. I'm curious at a university level or even at the postgraduate level.
1.- There exists an axiomatic construction of this subject?
2.- What is taught in this subject or what is the aim of the subject?
3.- There exists good books to self-learn about combinatorics?
4.- Is it important to a mathematical researcher to know about this or is it just for fun?
Thanks in advance.
Questions 2, 3, and 4 are way too broad (and you should be able to find your answer to 3 by searching this site, which you should be doing before posting questions).
As far as I know, combinatorics is never approached axiomatically. Here are the only reasons I can think of to give special attention to the axioms of a theory
If lots of paradoxes are cropping up, then you should take care to define the axioms. For example, we needed ZF(C) to solve the problem of Bertrand's paradox, and most probability "paradoxes" (especially Borel's paradox and the two-envelope problem) dissipate when formalized using Kolmogorov's axioms. This is not just not a problem in combinatorics.
Having a list of axioms defining a theory makes it easy to apply that theory in lots of different situations. For example, if you want to apply homology theory to something, you just need to check the Eilenberg-Steenrod axioms. Combinatorics is always applied in a obvious way, so this is no problem.
You want to consider what happens when you alter the axioms. The biggest example here is geometry with the parallel postulate, but there are many others. I have never heard of such a thing being done in combinatorics. Everything is finite, so there is really only one sensible way to do things.