Comment by qsort
I find the presentation a bit confusing and I'm already very familiar with this material.
From a purely mathematical point of view, why this choice of topics? You correctly point out that counting partitions has no closed formula, but there are a lot of related problems (Sitrling numbers of the two kinds for example) which are of more practical utility (e.g. they are related to sums of powers formulas). If that's too advanced for your audience then why not present more standard tricks like combinations with replacement aka stars and bars?
From a programmer's point of view, you could have focused more on how to generate subsets, permutations, partitions etc. in a memory-efficient way, for example how the Python stdlib does it.
Also, the factorial number system and the binomial base of univariate polynomials are definitely not "alternatives to base 2 in computer architectures".
Don't take this the wrong way but I struggle to see who you are writing for.
I am just a simple web developer and I found this stuff interesting. Learning about array programming has taught me the importance of combinatorics and its possible application to GPU programming. It is just the first chapter in a series of articles, I'd say let the man cook.