We'll just finish up with a couple of exercises that people might do to check

their understanding of this lecture. this one is a note from the book where it

talks about using these techniques to enumerate what's called a Super Necklace

which is labelled cycle of cycles. and you can look in the book for a

definition of that and see how the symbolic transfer gives this generating

function. and the problem is to show that that's

asymptotic to 1 over n times 1 minus 1 over e to the minus n.

so and there's a hint take derivatives. so that's we'll do lots more applications

of the meromorphic transfer theorem later on.

And then this is just an exercise on a different kind of combinatorial object.

so, again, the best thing to do, particularilly for people without much

familiarity with complex analysis is to review the lecture in the lecture notes

and also to look at the book in the section on complex analysis.

As usual, there is much much more in the book than I have a chance to talk about

in lecture. So your best strategy for now is to try

to get a feeling for what's there, pursue what you find interesting but you're not

going to be able to understand every detail without putting in many, many

hours. it's, our, our, our goal is to give you

some familiarity with the major concepts and give you a place to research later on

when you discover problems that you know, analytic combinatorics might be effective

for. and go ahead and write up a solution to

that note. and then also, I'd urge people to do

these simple programming exercises just to connect the abstractions that I've

talked about a little bit closer to the real world.

So, one thing you can do is compute the percentage of permutations say, that have

no singleton or doubleton cycles. and you can make a recurrence to go ahead

and compute that. And then compare it with the asymptotic

estimate that we get from the analytic combinatorial, save for n equals 10 and

and 20. another thing you do is get like code on

the book's site for plotting complex numbers on a greyscale image according to

the absolute value. And and, and plot the function associated

with that in a Super Necklace, just or plot some other function that you find

interesting. just to see how easy it is to create

these kinds of images. because we're going to be them to gain

intuition about the kind of functions that we talk about later on.

so, again next time we'll look at, lots, lots, lots and lots more applications of,

of, this basic transfer theorem.