All right. Then let's finish up with a few exercises

that you might try to check your understanding of the material in this

lecture. so the first one is an exercise that has

to do with calculating leaves in Cayley Trees which you recall are unordered

rooted trees and shows it's, it's a node which gives much of the results and you

have to fill in some of the details. but that's a good exercise to check the

use of bivariate generated functions and symbolic methods for a useful problem.

the second one is recreational mathematics.

so if you take all the numbers that are formed the decimal numbers, integers, and

digit 1 is used once, digit 2 is used twice, and so forth, digit 9 used nine

times. so that's 45-digit numbers.

And you take all of them and add them together.

if you use if you do that with a extended precision arithmetic package you'll find

out that you have a huge number of nines in a row, and actually lots of other

nines. and so you're supposed to explain that.

and and so this was actually discovered in 1150 AD and so maybe with ana-,

analytic combinatorics we can better understand it.

and so, read again the chapter in the text about multivariate generating

functions. And again, there's quite a bit of

advanced material in there. It's not attended that everybody read

every page but rather to get a good idea of what's in there and read more detail

about what you find interesting. write up solutions to those 2 exercises

and try the programming exercise. Again, just to validate the math.

generate some random permutations and check that the number of cycles that you

get in those permutations is, is about HN.

even, even for Math it's worthwhile to validate the Math that we do to better

understand the der-, derivations. so those are the exercises for or

combinatorial parameters lecture.