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We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently:…
We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, ie…
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…
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…
We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…
Let \sum_{n\in N^d} f_{n_1, ..., n_d} x_1^{n_1}... x_d^{n_d} be a multivariate generating function that converges in a neighborhood of the origin of C^d. We present a new, multivariate method for computing the asymptotics of the diagonal…
The field of analytic combinatorics in several variables (ACSV) develops techniques to compute the asymptotic behaviour of multivariate sequences from analytic properties of their generating functions. When the generating function under…
We provide a complete combinatorial and asymptotic analysis of positive linear systems of equations in one catalytic variable that appear in several combinatorial problems such as in lattice path counting or stack-sortable permutation…
Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the…
A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's…
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.
Uniform asymptotic formulae for arrays of complex numbers of the form $(f_{r,s})$, with $r$ and $s$ nonnegative integers, are provided as $r$ and $s$ converge to infinity at a comparable rate. Our analysis is restricted to the case in which…
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two…
The composition $\mathcal{F} \circ \mathcal{G}$ of two combinatorial classes $\mathcal{F}$ and $\mathcal{G}$ is a standard combinatorial construction and translates into the composition $F(G(z))$ of their corresponding counting generating…
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…
For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length…
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…
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many…
We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized…
We investigate depoissonization, the problem of recovering asymptotics of sequence coefficients from their exponential generating function. Classical approaches rely on complex-analytic growth conditions, but here we develop real-variable…