Advances in Combinatorial Mathematics

Proceedings of the Waterloo Workshop in Computer Algebra 2008
1. Auflage 2009
ISBN: 978-3-642-03562-3
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 174 Seiten

The Second Waterloo Workshop on Computer Algebra was dedicated to the 70th birthday of combinatorics pioneer Georgy Egorychev. This book of formally-refereed papers submitted after that workshop covers topics closely related to Egorychev's influential works.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Foreword;6
2;Preface;8
3;Contents;10
4;Method of Coefficients: an algebraic characterization and recent applications;11
4.1;Introduction;11
4.2;The method of generating functions as a method of summation (the method of coefficients);12
4.2.1;Computational scheme;12
4.2.2;Operations with formal power series and the inference rules;13
4.2.3;The problem of completeness;16
4.2.4;Connection with the theory of analytic functions;19
4.3;Several recent applications;19
4.3.1;The characteristic function of the stopping height for the Collatz conjecture;19
4.3.2;Computation of combinatorial sums in the theory of integral representations in Cn;24
4.3.3;Combinatorial computations related to the inversion of a system of two power series in Cn;27
4.4;Algebraic characterization of the method of coefficients as a method of summation;29
4.5;References;36
5;Partitions With Distinct Evens;41
5.1;Introduction;41
5.2;Proof of Theorem 2.2;44
5.3;Proof of Theorem 2.3;45
5.4;Proof of Theorem 2.6;46
5.5;Conclusion;47
5.6;References;47
6;A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings;48
6.1;Introduction;48
6.2;Classical group characters;50
6.3;Auxiliary identities;52
6.4;Proofs of theorems;55
6.5;Combinatorial interpretations;59
6.6;More factorization theorems;66
6.7;References;68
7;On multivariate Newton-like inequalities;69
7.1;Introduction;69
7.2;Univariate Newton-like Inequalities;72
7.2.1;Propagatable sequences (weights);72
7.3;Multivariate Case;74
7.3.1;Generalized van der Waerden-Egorychev-Falikman lower bounds;75
7.3.2;General monomials;78
7.3.3;A lower bound on the inner products of H-Stable polynomials;80
7.4;Multivariate Newton Inequalities;81
7.5;Comments and open problems;83
7.6;References;85
8;Niceness theorems;87
8.1;Introduction and statement of the problems;87
8.2;Examples;89
8.2.1;Lots of compatible structure examples;89
8.2.2;Universal object examples;97
8.2.3;Niceness theorems for Hopf algebras;100
8.2.4;Large vs nice;101
8.2.5;Extremal objects and niceness;102
8.2.6;Uniqueness and rigidity and niceness;103
8.2.7;Counterexamples and paradoxical objects;103
8.2.8;An excursion into formal group theory;106
8.2.9;The amazing Witt vectors and their gracious applications;110
8.2.10;The star example: Symm;115
8.2.11;Product formulas;117
8.3;Some first results and theorems;117
8.3.1;Freeness theorems;117
8.3.2;On the Lazard universal formal group theorem;118
8.3.3;Objects and isomorphisms in connection with Symm;119
8.4;References;127
9;Method of Generating Differentials;134
9.1;Introduction;134
9.2;Variables;137
9.2.1;Free Abelian Groups;137
9.2.2;Polynomial Rings;138
9.2.3;Power Series Rings;138
9.2.4;Fields of Generalized Power Series;139
9.3;Differentials;140
9.3.1;Kähler Differentials;140
9.3.2;Finite Differentials;141
9.3.3;Differentials for Generalized Power Series;141
9.4;Residues;143
9.4.1;Local Cohomology Residues;143
9.4.2;Logarithmic Residues ;144
9.5;Implementations;147
9.5.1;Inverting Combinatorial Sums;147
9.5.2;Compositional Inverses and Lagrange Inversions;148
9.5.3;MacMahon's Master Theorem;150
9.5.4;Dyson's conjecture;152
9.5.5;Constraints of Analytic Functions;155
9.6;References;160
10;Henrici's Friendly Monster Identity Revisited;162
10.1;Introduction ;162
10.2;Egorychev's Method in Action ;163
10.2.1;Reduction to a single sum ;164
10.2.2;Simplifying the single sum ;165
10.3;MultiSum in Action ;166
10.4;GeneratingFunctions in Action ;167
10.5;Conclusion ;168
10.6;References;169
11;The Automatic Central Limit Theorems Generator (and Much More!);171
11.1;References;180