All right, so let's summarize what we've done in this lecture.

as promised, we've looked at from an analytic combinatorics point of view a

whole range of classic combinatorial classes.

from bit strings to derangements, to surjections, to alignments, to set

partitions, integers, compositions. In every single one of 'em, we took a

specification, immediately to a generating function with the, via the

symbolic method, and then, immediately to coefficient asymptotics, using the

meromorphic transfer theorem. Not only that, not only are we able to do

these classic combinatorial classes but also we can do variations.

Like restrict the bit strings to not contain a particular pattern.

Or generalize derangements where We parameterize the number of cycles that

are disallowed. same way with the surjections and for

compositions all different types of restrictions and as well as the

numerants. all of those things, again, are the same

basic idea of a speck that goes into a generating function, goes to coefficient

asymptotics immediately. Handling these types of problems without

analytic combinatorics, which you can find in the literature is possible

because underlying the calculations, there's nothing particularly novel or

important to this point. it's possible, but it requires a huge

amount of calculation. And analytic combinatorics allows us to

skip over the top. to immediately derive asymptotic results

without much calculation. and that's important because the idea of

a speck gives us an open ended way to specify combinatorial classes.

without having to worry that we may not be able to understand their coefficient

asymptotics. actually we can for this huge set of

combinatorial classes. so as promised I, I think in this lecture

you've seen that our transfer theorem for mero, meromorphic generating functions

gives immediate analysis of a large variety of classes.

We can just as easily handle variations. And then, we have the idea of a schema,

which unifies the analysis for a whole family of classes, including analysis of

parameters. now and also there have been several

other schemas developed that are talked about in the text.

so what we have do next is look at generating functions that are not

meromorphic. and that's going to be the topic of the

next two lectures when we get functions with singularities that are not isolated

that are not poles. and that'll be our topic for the next two

lectures.