Now let's finish up with a quick summary of the things that we've covered in this lecture. the main topic has been the symbolic method for labelled classes, and particularly the transfer theorem shows how the constructions that we consider transfer to operations on generating functions. So we consider the disjoint union operation. And the main one, which is the labelled product operation, where we, we relabel objects in all possible ways. And we would get a product on the generating function, sequences of length k, which we would generate in function a to the z to the k, sequence of any length 1 over 1 minus. Same with sets of length k or sets of any size, and cycles of length k or cycles of any size. And there's many others that people have defined, like the box product which has a corresponding operation on the generating function. with these basic operations and just a few examples, there's many, many other examples in the book. we were able to define a whole host of commonatorial classes. all with a relatively simple description and then leading to Directly from the transfer theorem to exponential equations that exponential generating functions have to satisfy. And we have variations on these to lead to a, a huge variety of different common notarial classes and still have a specific equation that the generating functions have to satisfy. That fits with our overview to analyze properties of a large combinatorial structures, we use the symbolic method to get directly generating function equation. now its, important to note that the equations that we get vary widely in nature. Some of them are quite simple, some of them are quite complicated. but those are the generating functions of the counting sequences that we seek. so in the second part where we derive asymptotics from generating functions, we're also going to need general tools that can handle all of these different types, of generating functions. But still, the derivations that give us the generating functions, were all, really, essentially, quite simple. in fact to the point where again this brings in also unlabeled classes. We we can automate the transfer from specifications to generating functions. and in fact Philippe and, and his students Bruno Salvy and Paul Zimmerman did so in the 90's. in much of this work is reflected in modern symbol manipulation systems. Another thing we can do is use specs to generate huge random structures. one easy approach is to just take the spec and use a recursive program that is based on the spec. and that's okay but it doesn't get you really large structures because it takes quadratic time. but there is a probabilistic method that gets for a given size, that gets a random structure of, approximately, that size and works in linear time. And this is, becomes, increasingly important, for checking, whether a combinatorial model that someone might have of the real world, actually matches with observed in the real world. Can generate, a large number of random structures and check if they're properties correspond to the real world thing, object that's being studied. whether it be a DNA sequence or a computer network or whatever else you've found that people are studying nowadays. so that's called Bolton sampling in that's reference for for that as well. but I think again it's all about the utility of generating functions. and really the bottom like is that Without something like the symbolic method or generating functions. people were doing really horrendously detailed and difficult calculations. and this approach really does eliminate virtually all of them. That's the summary of the symbolic method both for labelled and unlabelled methods.
