3,971 research outputs found
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Terminal chords in connected chord diagrams
Rooted connected chord diagrams form a nice class of combinatorial objects.
Recently they were shown to index solutions to certain Dyson-Schwinger
equations in quantum field theory. Key to this indexing role are certain
special chords which are called terminal chords. Terminal chords provide a
number of combinatorially interesting parameters on rooted connected chord
diagrams which have not been studied previously. Understanding these parameters
better has implications for quantum field theory.
Specifically, we show that the distributions of the number of terminal chords
and the number of adjacent terminal chords are asymptotically Gaussian with
logarithmic means, and we prove that the average index of the first terminal
chord is . Furthermore, we obtain a method to determine any next-to
leading log expansion of the solution to these Dyson-Schwinger equations, and
have asymptotic information about the coefficients of the log expansions.Comment: 25 page
A Generalization of the Eulerian Numbers
In the present paper we generalize the Eulerian numbers (also of the second
and third orders). The generalization is connected with an autonomous
first-order differential equation, solutions of which are used to obtain
integral representations of some numbers, including the Bernoulli numbers.Comment: 13 pages, preprin
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
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
On the non-holonomic character of logarithms, powers, and the n-th prime function
We establish that the sequences formed by logarithms and by "fractional"
powers of integers, as well as the sequence of prime numbers, are
non-holonomic, thereby answering three open problems of Gerhold [Electronic
Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex
analysis, namely a conjunction of the Structure Theorem for singularities of
solutions to linear differential equations and of an Abelian theorem. A brief
discussion is offered regarding the scope of singularity-based methods and
several naturally occurring sequences are proved to be non-holonomic.Comment: 13 page
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