Related papers: A SageMath Package for Analytic Combinatorics in S…
We introduce the new sage_acsv package for the SageMath computer algebra system, allowing users to rigorously compute asymptotics for a large variety of multivariate sequences with rational generating functions. Using Sage's support for…
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…
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…
The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several…
Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several…
We present a strategy for computing asymptotics of coefficients of $d$-variate algebraic generating functions. Using known constructions, we embed the coefficient array into an array represented by a rational generating functions in $d+1$…
Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that…
In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko…
Let F be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate…
We consider asymptotics of power series coefficients of rational functions of the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of…
In recent years, the techniques of analytic combinatorics in several variables (ACSV) have been applied to determine asymptotics for several families of lattice path models restricted to the orthant $\mathbb{N}^d$ and defined by step sets…
Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…
Making use of a newly developed package in the computer mathematics system SageMath, we show how to perform a full asymptotic analysis of certain types of sums that occur frequently in combinatorics, including explicit error bounds. We…
We combine tools from homotopy continuation solvers with the methods of analytic combinatorics in several variables to give the first practical algorithm and implementation for the asymptotics of multivariate rational generating functions…
This article establishes an asymptotic theory for volatility estimation in an infinite-dimensional setting. We consider mild solutions of semilinear stochastic partial differential equations and derive a stable central limit theorem for the…
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…
We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken…
Existing methods of series analysis are largely designed to analyse the structure of algebraic singularities. Functions with such singularities have their $n^{th}$ coefficient behaving asymptotically as $A \cdot \mu^n \cdot n^g.$ Recently,…
We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the…