Multivariate Limit Theorems and Algebraic Generating Functions

Abstract

The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of Analytic Combinatorics in Several Variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. This thesis provides results for two areas of ACSV: limit theorems and asymptotics of algebraic generating functions. For both, the aim is to provide readers a blueprint to apply the powerful tools of ACSV in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields. First, we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large class of combinatorial generating functions. To illustrate the techniques involved, we also establish explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem conjectured in work of Chung et al.), integer compositions, and n-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in an arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate LU-factorization. The second part of this thesis shifts focus to the calculation of asymptotics of multivariate algebraic generating functions through ACSV. So far, the methods of ACSV have largely focused on rational (or, more generally, meromorphic) generating functions, although many natural combinatorial objects have generating functions with algebraic singularities. In this part, we survey techniques for analyzing multivariate algebraic generating functions, going into detail specifically for the process of embedding an algebraic generating function into a sub-series of a rational function of more variables. Other methods mentioned include explicit singularity analysis of algebraic singularities, and manipulation of complex integrals over algebraic hypersurfaces. We give implementations of the embedding techniques in the SageMath computer algebra system, and provide examples from the combinatorics literature.