Just to summarize we've looked at many different applications of singularity analysis starting with some familiar ones rooted ordered trees and binary trees. we looked at the unary-binary trees and Cayley trees always getting generating function by the symbolic method and coefficient asymptotics. By a transfer theorem built from singularity analysis. we looked at properties of mappings. 2-regular graphs, labeled a higher piece, and other implicit, tree-like classes, and always being able to go from construction to generating function to coefficient asymptotics. And these are only representative examples for each one of them, and there's many, many other types of classes that are similar that we can define and still get immediately to the coefficient asymptotics. we have a very powerful and general calculus for deriving estimates of, of coefficients from combinatorial constructions. if you can specify it, you can analyze it. Singularity analysis is a very effective approach. For generating function equations for classes with the GFs that are not. It actually, that are not meromorphic. It actually also works for meromorphic functions it's, it's more general. And some of these schemas that we've actually done were meromorphic. Though we didn't, we just didn't we just didn't talk about it. Could be. so the idea is that what, we have these schema that can unify the analysis for entire families of classes. we talked about the x blog, and simple variety of trees, and the context free classes last time, and we didn't show any applications this time, in implicit tree like classes, and we can always get out to the a coefficient asymptotics for large families of combinatorial classes. and there's other schema also that have been proven. Now, not every example goes as smoothly as the ones that I've shown in lecture. and particularly for context free and related classes the calculations involved to actually find the specific constant can be complicated. but usually it could be mechanized and it's reasonable to expect for a great variety of classes that you're going to find this kind of behavior just using the basic principle of coefficient asymptotics is the location of the singularity is going to give you the asymptotic growth and the nature of the singularity is going to tell you about the sub-exponential factor in the constant. And in a great many cases we can compute these things exactly into arbitrary asymptotic accuracy. so next lecture we're going to talk about what to do with generating functions that have no singularities. but, singularity analysis is really at the heart of analytic combinatorics.
