Asymptotic Enumeration Methods

A remark on asymptotic enumeration of highest weights in tensor powers of a representation

We consider the semigroup $S$ of highest weights appearing in tensor powers $V^{otimes k}$ of a finite dimensional representation $V$ of a connected reductive group. We describe the cone generated by $S$ as the cone over the weight polytope of $V$ intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in $V^{otime...

Asymptotic Enumeration Methods

Airy Phenomena and Analytic Combinatorics of Connected Graphs

Until now, the enumeration of connected graphs has been dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enum...

Enumerating Regular Expressions and Their Languages

In this chapter we discuss the problem of enumerating distinct regular expressions bysize and the regular languages they represent. We discuss various notions of the size of a regularexpression that appear in the literature and their advantages and disadvantages. We consider aformal definition of regular expressions using a context-free grammar.We then show how to enumerate stri...

Pólya's Random Walk Theorem

This note presents a proof of Pólya’s random walk theorem using classical methods from special function theory and asymptotic analysis.