Review Theorem I.1 in AC if you don’t remember the basic constructions.
Appendix A.1 and A.4 in AC. This goes over the cycle construction, but leaves
much to the reader. You don’t necessarily need to follow every step, but it
would be good to have an idea of where this is going.
Optionally take a look at the FLINT docs. I plan to
go over the core stuff of interest though.
Agenda:
Review basic definitions of combinatorial classes and fundamental
constructions.
Derive the cycle construction, which we skipped when originally visiting
Theorem I.1. (Neil will be doing this one.)
Efficient extraction of high coefficients using the FLINT library (we’ll
probably use the python-flint wrapper as it’s much more practical for
exploration).
Part of the efficiency comes from the built-in FLINT numerical techniques,
but OGF representation still matters a lot for performance. We’ll go over
some pathological examples and show how to rewrite them to make them
faster.
Rule of thumb: wherever possible, simplify the GF symbolically and define
the whole thing in-line. Avoid delaying/abstracting functions and in
particular avoid composing power series.
Notes:
Spent longer than anticipated on the basic constructions and definitions. This
has motivated me to add a dedicated section to the site that covers this so
people can refer to it. (To come.)
We got maybe a third of the way through the cycle construction, mostly
covering the basics of the Möbius function and Dirichlet generating functions
and their relationship to Euler totient (φ) and Riemann ζ functions. We’ll
need to split the rest of it into separate sessions.
Neil posted an
expanded write-up of Appendix A.1 and
A.4. It fills in some detail left out of AC.
We didn’t get to the FLINT/numerical methods at all. I’m considering doing
that in a dedicated session.