Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

TL;DR

This paper introduces effective algorithms for analytic combinatorics in several variables, providing new asymptotic methods, complexity results, and applications to lattice path enumeration, including solving open conjectures.

Contribution

It develops rigorous algorithms for ACSV, offers the first complexity analysis, and applies these methods to solve open problems in lattice walk enumeration.

Findings

Derived asymptotics for lattice walks in restricted regions.

Proved several open conjectures on lattice walk asymptotics.

Analyzed lattice walk models with weighted steps.

Abstract

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.