12,798 research outputs found
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
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
Counting Gauge Invariants: the Plethystic Program
We propose a programme for systematically counting the single and multi-trace
gauge invariant operators of a gauge theory. Key to this is the plethystic
function. We expound in detail the power of this plethystic programme for
world-volume quiver gauge theories of D-branes probing Calabi-Yau
singularities, an illustrative case to which the programme is not limited,
though in which a full intimate web of relations between the geometry and the
gauge theory manifests herself. We can also use generalisations of
Hardy-Ramanujan to compute the entropy of gauge theories from the plethystic
exponential. In due course, we also touch upon fascinating connections to Young
Tableaux, Hilbert schemes and the MacMahon Conjecture.Comment: 51 pages, 2 figures; refs updated, typos correcte
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
Staircase polygons: moments of diagonal lengths and column heights
We consider staircase polygons, counted by perimeter and sums of k-th powers
of their diagonal lengths, k being a positive integer. We derive limit
distributions for these parameters in the limit of large perimeter and compare
the results to Monte-Carlo simulations of self-avoiding polygons. We also
analyse staircase polygons, counted by width and sums of powers of their column
heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity:
An International Workshop On Statistical Mechanics And Combinatorics, 10-15
July 2005, Queensland, Australi
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