A new solution to Riordan’s problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated.

Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type

This thesis addresses a number of enumerative problems that arise in the context of quantum field theory and in the process of renormalization. In particular, the enumeration of rooted connected chord diagrams is further studied and new applications in quenched QED and Yukawa theories are introduced. Chord diagrams appear in quantum field theory in the context of Dyson-Schwinger equations, where, according to recent results, they are used to express the solutions. In another direction, we study the action of point field diffeomorphisms on a free theory. We give a new proof of a vanishing phenomenon for tree-level amplitudes of the transformed theories.

On the Enumerative Structures in Quantum Field Theory

This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, is introduced. The Hopf algebraic structure on graphs permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories. The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together both structures provide an algebraic formulation of large graphs with constraints on the allowed subgraphs. These structures are motivated by and used to analyze renormalized zero-dimensional quantum field theory at high orders in perturbation theory. As a pure application of the Hopf algebra structure, an Hopf algebraic interpretation of the Legendre transformation in quantum field theory is given. The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems from combinatorics: The asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined in an application to zero-dimensional quantum field theory. Various quantities are explicitly given asymptotically in the zero-dimensional version of φ3, φ4, QED, quenched QED and Yukawa theory with their all order asymptotic expansions.

Graphs in perturbation theory

We derive a functional relation between the generating functions of connected chord diagrams and 2-connected chord diagrams. This relation enables us to calculate an asymptotic expansion for the number of 2-connected chord diagrams on $n$ chords. The asymptotic information obtained from this expansion refines the last established results and provides a simple alternative for calculating the asymptotic behaviour of certain Green functions in Quenched QED and Yukawa theory in the context of quantum field theory.

An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring is also closed under composition and inversion of power series. An `asymptotic derivation' is defined which maps a power series to the asymptotic expansion of its coefficients. Product and chain rules for this derivation are deduced. With these rules asymptotic expansions of the coefficients of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.

/pdf/generating-asymptotics-for-factorially-divergent-sequences-2co39qwuyg.pdf

Generating asymptotics for factorially divergent sequences

Exact Asymptotics of Meander Numbers.- Algorithms of Integral Representation of Combinatorial Sums and Their Applications.- Quasi-Symmetric Functions.- Equations over Free and Fully Residually Free Groups.- Recurring Sequences.- On Modules Supported in the Nullcone.- Towards Topological Classification of Univariate Complex Polynomials.- Applicability of Zeilberger's Algorithm to Rational Functions.- Bases explicites et conjecture n!.- A Generalized Cover Time for Random Walks on Graphs.- On the Square Root of Languages.- A General Cyclic Lemma for Multiset Permutation Inversions.- Enumeration of Some Labelled Trees.- On Aperiodic and Star-free Formal Power Series in Partially Commuting Variables.- Some Generalizations of Quasi-symmetric Functions and Noncommutative Symmetric Functions.- The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene.- Analytic Combinatorics of Chord Diagrams.- Specifying 2-trees.- About the Dynamics of Some Systems Based on Integer Partitions and Compositions.- Hecke Algebras of Type A at q = 0 and Quasi-differential Operators.- Central Characters and Conjugacy Classes of the Symmetric Group.- The Furedi-Hajnal Conjecture Implies the Stanley-Wilf Conjecture.- Generalized Integer Partitions, Tilings of Zonotopes and Lattices.- Topology of Colors.- The Number of Subgroups in the Fundamental Groups of Some Non-orientable 3-Manifolds.- Unranking of Labelled Combinatorial Structures.- On Gelfand-Tsetlin Bases for Representations of Classical Lie Algebras.- A Common Asymptotic Behavior for Different Classes of Sparse Labelled Graphs with Given Number of Vertices and Edges.- On Functions Which are Limits of Domino Tilings.- Jeux de tableaux.- The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes.- Special Formal Series Solutions of Linear Ordinary Differential Equations.- New Expressions for Level-restricted Kostka Polynomials.- A Combinatorial Interpretation of the Quotient-Difference Algorithm.- Regular Unicellular Dessins d'Enfants and Weil Curves.- Quantum Division Ring.- On the Integer Models of Plane Trees.- Congruences Compatible with the Shuffle Product.- Arithmetical Properties of [pk]-Series Associated to p-Typical Formal Group Laws and Applications.- A Note on Linear Preservers of a Certain Matrix Partial Order.- Trees of Diameter 4.- The Makar-Limanov's Construction of Algebraically Closed Skew Field via Mal'cev-Newmann Series.- About Nilpotency of Engel Algebras.- Belyi Functions Related to Plane Graphs: Multiplicities and Parasitic Solutions.- Construction of the Annihilator of a Linear Recurring Sequence over Finite Module with the help of the Berlekamp-Massey Algorithm.- Polylinear Recurring Sequences over a Bimodule.- An Improved Version of Standard Bases.- Effective Matrix Methods in Commutative Domains.- Polylinear Shift Registers and Standard Bases.- Burnside Matrices and Subgroup Embeddings in Finite Groups.- Distribution of Elements on Cycles of Linear Recurrences over Galois Field.- On Witt's Formula and Invariants for Free Lie Superalgebras.- Grobner-Shirshov Bases for Kac-Moody Algebras An(1) and Bn(1).- On Some Classes of Linear Representable Matroids.- On a Class of Families of Belyi Functions.- Finite Basis Property of S-Ideals of Finite Dimensional Forms.- Peak Reduction Technique in Commutative Algebra.- On Algebraic Degrees and Multiplicities for Parameters of Some Finite Group Rings.- Some Ring and Module Properties of Skew Power Series.- Some Ring and Module Properties of Skew Laurent Series.- Extending a Partially Ordered Set: Links with its Lattice of Ideals.- From Rational Functions to Regular Languages.- Proving a Class of Hypergeometric Identities by Means of Generating Functions.- Series generatrices exponentielles pour les ECO-systemes signes.- New Statistics on Non-crossing Trees.- Expected Number of Inversions After a Sequence of Random Adjacent Transpositions.- The Variety of Associative Rings, Which Satisfy the Identity x32 = 0, is Not Specht.- Polynomials of Bounded Tree Width.- Permutations Containing and Avoiding Certain Patterns.- On the Center of the Quantum Pseudoplactic Algebra.- ECO-Approximation of Algebraic Functions.- Methods for Computing the Concurrency Degree of Commutation Monoids.- An Application of Dumont's Statistic.- On Wohlfahrt Formula for Finite Abelian Groups.- Induction and Auslander-Reiten Sequences Over Crossed Products.- On Acceleration of Multiprecision Computation of Products and Sums of Products of Rational Numbers.- Combinatorial Aspects of Capelli Identities and Structure of Algebras.- Generators and the Template Method Pattern. A Paradigm for Combinatorial Experimentation.- A Software to Solve Strip Tiling Problems.- Author Index.

Formal Power Series and Algebraic Combinatorics : 12th International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings

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Formal Power Series and Algebraic Combinatorics : 12th International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings