61,212 research outputs found
Singularity analysis of generating functions
Résumé disponible dans les fichiers attaché
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
Singularity analysis, Hadamard products, and tree recurrences
We present a toolbox for extracting asymptotic information on the
coefficients of combinatorial generating functions. This toolbox notably
includes a treatment of the effect of Hadamard products on singularities in the
context of the complex Tauberian technique known as singularity analysis. As a
consequence, it becomes possible to unify the analysis of a number of
divide-and-conquer algorithms, or equivalently random tree models, including
several classical methods for sorting, searching, and dynamically managing
equivalence relationsComment: 47 pages. Submitted for publicatio
The relation between tree size complexity and probability for Boolean functions generated by uniform random trees
We consider a probability distribution on the set of Boolean functions in n
variables which is induced by random Boolean expressions. Such an expression is
a random rooted plane tree where the internal vertices are labelled with
connectives And and OR and the leaves are labelled with variables or negated
variables. We study limiting distribution when the tree size tends to infinity
and derive a relation between the tree size complexity and the probability of a
function. This is done by first expressing trees representing a particular
function as expansions of minimal trees representing this function and then
computing the probabilities by means of combinatorial counting arguments
relying on generating functions and singularity analysis
Tail generating functions for Markov branching processes
We give a concise self-contained presentation of known and new limit theorems
for the one-type Markov branching processes with continuous time. The new
streamlined proofs are based on what we call, the tail generating function
approach. Our analysis focuses on the singularity points of the master integral
equation for the probability generating functions of the current population
size
A repertoire for additive functionals of uniformly distributed m-ary search trees
Using recent results on singularity analysis for Hadamard products of
generating functions, we obtain the limiting distributions for additive
functionals on -ary search trees on keys with toll sequence (i)
with ( and correspond roughly
to the space requirement and total path length, respectively); (ii) , which corresponds to the so-called shape functional; and (iii)
, which corresponds to the number of leaves.Comment: 26 pages; v2 expands on the introduction by comparing the results
with other probability model
- âŠ