Asymptotics of multivariate sequences, part III: quadratic points

TL;DR

This paper develops methods to compute asymptotics of multivariate generating functions with quadratic singularities, applying the results to combinatorial problems like tilings and permutations.

Contribution

It introduces a novel approach combining topological deformations and Fourier-Laplace transforms to analyze quadratic singularities in generating functions.

Findings

Derived asymptotic formulas for coefficients near quadratic singularities

Applied methods to Aztec diamond tilings and cube groves

Provided new insights into combinatorial enumeration techniques

Abstract

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations.