25,277 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
Lattice paths of slope 2/5
We analyze some enumerative and asymptotic properties of Dyck paths under a
line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet
lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in
June 2014.Our approach relies on the work of Banderier and Flajolet for
asymptotics and enumeration of directed lattice paths. A key ingredient in the
proof is the generalization of an old trick of Knuth himself (for enumerating
permutations sortable by a stack),promoted by Flajolet and others as the
"kernel method". All the corresponding generating functions are algebraic,and
they offer some new combinatorial identities, which can be also tackled in the
A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar
results for other slopes than 2/5, an interesting case being e.g. Dyck paths
below the slope 2/3, which corresponds to the so called Duchon's club model.Comment: Robert Sedgewick and Mark Daniel Ward. Analytic Algorithmics and
Combinatorics (ANALCO)2015, Jan 2015, San Diego, United States. SIAM, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO), eISBN 978-1-61197-376-1, pp.105-113, 2015, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO
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
Complex bounds for multimodal maps: bounded combinatorics
We proved the so called complex bounds for multimodal, infinitely
renormalizable analytic maps with bounded combinatorics: deep renormalizations
have polynomial-like extensions with definite modulus. The complex bounds is
the first step to extend the renormalization theory of unimodal maps to
multimodal maps.Comment: 20 pages, 3 figure
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
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