Analytic Combinatorics – A Worked Example

There's an interesting extension to Analytic Combinatorics by Flajolet/Sedgewick called "Analytic Combinatorics in Several Variables" by Pemantle and Wilson: https://www2.math.upenn.edu/~pemantle/papers/ACSV.pdf and https://acsvproject.com/

It extends the original framework with a lot of useful new primitives, like Hadamard products. Super useful!

Thanks!

While this interesting, what are its main applications?

I'm currently using it to analyze a version of the Kaczmarz algorithm, generalizing my answer here: https://mathoverflow.net/a/490506/5429

But more generally, why do we want to analyze combinatorics and algorithms? I suppose it gives us some confidence that we are actually making progress.