2,402 research outputs found
A Note on the generating function of p-Bernoulli numbers
We use analytic combinatorics to give a direct proof of the closed formula
for the generating function of -Bernoulli numbers.Comment: 6 page
Recursive Combinatorial Structures: Enumeration, Probabilistic Analysis and Random Generation
In a probabilistic context, the main data structures of computer science are viewed as random combinatorial objects.
Analytic Combinatorics, as described in the book by Flajolet and Sedgewick, provides a set of high-level tools for their probabilistic analysis.
Recursive combinatorial definitions lead to generating function equations from which efficient algorithms can be designed for enumeration, random generation and, to some extent, asymptotic analysis. With a focus on random generation, this tutorial first covers the basics of Analytic Combinatorics and then describes the idea of Boltzmann sampling and its realisation.
The tutorial addresses a broad TCS audience and no particular pre-knowledge on analytic combinatorics is expected
Asymptotics of multivariate sequences IV: generating functions with poles on a hyperplane arrangement
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 residues and saddle-point approximations. Because the
singular set of F is the union of hyperplanes, we are able to make explicit the
topological decompositions which arise in the multivariate singularity
analysis. In addition to effective and explicit asymptotic results, we provide
the first results on transitions between different asymptotic regimes, and
provide the first software package to verify and compute asymptotics in
non-smooth cases of analytic combinatorics in several variables. It is also our
hope that this paper will serve as an entry to the more advanced corners of
analytic combinatorics in several variables for combinatorialists
Analytic combinatorics for a certain well-ordered class of iterated exponential terms
International audienceThe aim of this paper is threefold: firstly, to explain a certain segment of ordinals in terms which are familiar to the analytic combinatorics community, secondly to state a great many of associated problems on resulting count functions and thirdly, to provide some weak asymptotic for the resulting count functions. We employ for simplicity Tauberian methods. The analytic combinatorics community is encouraged to provide (maybe in joint work) sharper results in future investigations
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
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 variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
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