What should I study after introductory Combinatorics?

Question

So, I am a high school student and my original plan was to major in Computer Science and Minor in mathematics. I have recently discovered an intense interest in mathematics and am contemplating a double major in Computer Science and Mathematics. I am currently taking Algebra 2 and plan to take pre-calculus over the summer so that I can be on track to take all the math courses my school offers. Besides that I have been approved to self-study discrete mathematics next year and am currently taking an online introductory course in Combinatorics. I'm pretty sure I will be finished with the Combinatorics course within the next month or so, which leads me to my question: What should I study next that would help complement knowledge in Combinatorics? Am I studying things in a non-beneficial order?

Answer 1

This is how I went through my Stats&Probability course in A-level (Class of 2017, H2 Math + H2 FMath, Singapore) which I think has a really nice flow to it:

1. Intro to combinatorics
2. Intro to probability
3. Discrete RV (Intro, Binom, Geo, Poisson)
4. Continuous RV (Intro, Normal, Uniform, Expo)
5. Poisson process (Poisson+Expo)
6. Hypothesis testing + Confidence Intervals (Z, T, Chi-squared)

You might want to master from the pure math side:

Before (3): Series & Sequence; Summation techniques; Recurrence Relations (1st&2nd order linear)

Before (4): Single-variable Calculus

Good to have but not really needed: Intro Linear Algebra

Answer 2

If you want to augment what you have learned in combinatorics then I would recommend:

1. Graph theory
2. Number theory
3. Probability
4. More advanced combinatorics.

For the last suggestion, I would recommend the book “Enumerative Combinatorics volume 1” by Richard P. Stanley. You can get it for free on the web.

Answer 3

First of all make sure that you got combinatorics on 100% because combinatorics itself is widely used in programming and is essential to learn probability which is also widely used in programming. So the best choice after combinatorics is definately probability. Probability is usualluy the topic that is taught after combinatorics in most textbooks. After probability, it would be really handful to learn Sequences and analysis. And if you are interested in neural networks Jump in Calculus.