This is the last lecture in the analytic combinatorics class, so it's appropriate to do a wrap up and a survey of the things that we've done. And I've used this slide at the beginning of every lecture to give the basic overview that what we're after is mechanisms whereby we can specify a combinatorial problem, and from that specification, immediately and automatically derive a generating function equation, and then use complex asymptotics to get asymptotic estimates of coefficients, which give, which give us the quantitative results, that we're looking for. the basic ideas, are I'll, I'll say in words because they're worth restating. first of all, we have combinatorial specifications that give us succinct ways to define discrete structures. And this is an extensible system of specifications and people are discovering new ones every day. Associated with the specifications, we have the symbolic method, which gives us a way to transform the specifications to equations that define generating functions. We're treating the generating functions as formal objects that we can immediately derive definitions from the specifications. Then we pivot and treat the generating functions as analytic objects, as functions in the complex plane. and we can get coefficients, estimates of the coefficients, by doing that. Either using Cauchy's coefficient formula directly when the singularities are poles, or using singularity analysis when we have essential singularities, or saddle-point asymptotics when there's no singularities at all. so, and more important, in many situations we have a schema that for, various combin, combinatorial classes that give universal asymptotic laws. So, in, in many cases where we don't, need to go down to the detail of asymptotics. Certain combinatorial classes, just by their structure, asymptotic laws, tell us quantitative bounds on, on their properties. so, early on, we looked at constructions and symbolic transfers for unlabeled objected and for labeled objects. and these basic constructions gave us the tools to specify many, many classic combinatorial problems and get associated generating functions immediately. and then, after treating generating functions as analytic objects functions in a complex plane we saw explicit ways to transfer from generating function equation to coefficients. for meromorphic functions, there's a general transfer theorem. if the generating function is in the standard scale, we saw at the beginning of the singularity analysis lecture that we could immediately transfer functions like that into coefficient asymptotics. if we have square root or logarithmic singularities, we can use singularity analysis. And if there's no singularities, we can use the saddle point method as discussed in this lecture. But more than that, there's schemas. We can organize. We saw in several lectures that we can organize combinatorial problems into broad schemas that cover, infinitely many combinatorial types and, and the simple asymptotic laws that govern them. so we looked at supercritical sequence schema. if it's built as a sequence with certain technical conditions, we know the coefficient asymptotics, or we look at f x blog. If it's a set under certain conditions, again, certain technical conditions we get asymptotic. Or a simple variety of trees, where it's recursive against certain technical conditions or context-free, or implicit tree-like classes. These schema cover a very, very broad variety of combinatorial constructions. And we immediately have coefficient asymptotics under certain technical conditions. One of the most important aspects of analytic combinatorics is the discovery of schemas like this and the associated universal laws that's the very essence of analytic combinatorics. There's plenty of other examples in the book. So from the beginning I've been repeating Philippe's mantra, if you can specify it you can analyze it. And I'll just click through the 20 or so examples from binary trees to surjections to compositions, alignments, compositions of various types, permutations with derangements order trees of two regular graphs. Lots of different types of combinatorial classes finally ending up with involutions and set partitions in this lecture. I noticed that that's just a small example for every one of these examples is variations, that again can be handled in a, in a uniform manner. so just in case someone asks you took analytic combinatorics, what is that? this is, the answer. it seems to enable precise quantitative predictions of the properties of large combinatorial structures. It's a theory that's emerged over recent decades as essential, not just for the analysis of algorithms but also for the study of scientific models in other disciplines, including statistical physics, computational biology and information theory. So, then the question is what's next. Well we've only really looked at the tip of the iceberg and this an introduction to analytic combinatorics. and there's quite a bit more in the book and in the research literature for further study. so we only covered the basic constructions for example. There's many associated many other constructions that people have studied in symbolic transfers that lead down paths through many other important applications. For example, there's a whole family of, there's quite a bit of discussion about paths and lattices and other types of combinatorial problems that we haven't talked about. Looking at the details of the singularity analyses proofs is really worthwhile to, that really is a watershed moment in Analytic Combinatorics because of the development of those transfers. there's a lot of conditions having to do with periodicity, irreducibility and what happens with algebraic functions and context-free languages that are fascinating mathematically and important in applications and definitely worth studying. And we have only a few moments in this course to talk about and other schemes as we move on we're going to talk about five or six at this point these other ones. The Drmota Llaley Woods theorem that helps us with that, exclusions of systems of generating functions equations, that's another key, an insightful piece of mathematics that's at the core of analytic combinatorics and definitely we're going to study. reading more about saddle point approximations and the associated technical conditions and the trade off between the central approximation and neglecting the tails. That's also important area to study in more detail. The primary reason for that is the role of both that kind of saddle point in limit laws and multivariate asymptotics but we're not looking just for numeration but also distributions of properties of combinatorial objects. and there are developed in that of the book are important and far-reaching and widely applicable. General laws that tell us that is often the case that parameters that come between objects again in simple asymptotic behavior. And then applications, there's many appliactions in the book and many applications in the research literature. And really its the applications that inform the mathematics that we develop in analytic combinatorics. So,, for an overview of whats going on recently and in also Fajolet's life's work in papers. I recorded a a postscript to this course I called if you can specify it you can analyze it the lasting legacy of Phillipe Flajolet. And that's the second part of, of a lecture, that I gave earlier this year, and this first part with a preview of this course so take a look at that if you're interested in further study in analytic combinatorics. and finally I want to finish by talking about other resources or seamless plugs or things that are available related to the information in this course. we've been talking mostly about the analytic combinatorics book prior to that, part one of the course was about analysis of algorithms. And then algorithms book joint with Kevin Wynne, also have online materials and a course. And if you're a mathematician and not familiar with programming, we have Introduction to Programming. and all of these have associated web resources that you can, take a look at, to get, lots more information or to engage, with, with the material. and also you're here because, of, of online courses, we have online courses associated with algorithms with analysis of algorithms, and with analytic, analytic combinatorics. And also as special treat for people, that are taking this course, there's a t-shirt, and if you go to the book site you can get details on ordering the t-shirt. so, just the a final though, and this is a Winston Churchill quote, it's also a rock anthem. but now this is not the end. It's not even the beginning of the end but it's perhaps the end of the beginning and that's the way I'd like you to think about analytic combinatorics.