Advances in Combinatorial Mathematics (eBook)

Foreword 6
Preface 8
Contents 10
Method of Coefficients: an algebraic characterization and recent applications 11
Introduction 11
The method of generating functions as a method of summation (the method of coefficients) 12
Computational scheme 12
Operations with formal power series and the inference rules 13
The problem of completeness 16
Connection with the theory of analytic functions 19
Several recent applications 19
The characteristic function of the stopping height for the Collatz conjecture 19
Computation of combinatorial sums in the theory of integral representations in Cn 24
Combinatorial computations related to the inversion of a system of two power series in Cn 27
Algebraic characterization of the method of coefficients as a method of summation 29
References 36
Partitions With Distinct Evens 41
Introduction 41
Proof of Theorem 2.2 44
Proof of Theorem 2.3 45
Proof of Theorem 2.6 46
Conclusion 47
References 47
A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings 48
Introduction 48
Classical group characters 50
Auxiliary identities 52
Proofs of theorems 55
Combinatorial interpretations 59
More factorization theorems 66
References 68
On multivariate Newton-like inequalities 69
Introduction 69
Univariate Newton-like Inequalities 72
Propagatable sequences (weights) 72
Multivariate Case 74
Generalized van der Waerden-Egorychev-Falikman lower bounds 75
General monomials 78
A lower bound on the inner products of H-Stable polynomials 80
Multivariate Newton Inequalities 81
Comments and open problems 83
References 85
Niceness theorems 87
Introduction and statement of the problems 87
Examples 89
Lots of compatible structure examples 89
Universal object examples 97
Niceness theorems for Hopf algebras 100
Large vs nice 101
Extremal objects and niceness 102
Uniqueness and rigidity and niceness 103
Counterexamples and paradoxical objects 103
An excursion into formal group theory 106
The amazing Witt vectors and their gracious applications 110
The star example: Symm 115
Product formulas 117
Some first results and theorems 117
Freeness theorems 117
On the Lazard universal formal group theorem 118
Objects and isomorphisms in connection with Symm 119
References 127
Method of Generating Differentials 134
Introduction 134
Variables 137
Free Abelian Groups 137
Polynomial Rings 138
Power Series Rings 138
Fields of Generalized Power Series 139
Differentials 140
Kähler Differentials 140
Finite Differentials 141
Differentials for Generalized Power Series 141
Residues 143
Local Cohomology Residues 143
Logarithmic Residues 144
Implementations 147
Inverting Combinatorial Sums 147
Compositional Inverses and Lagrange Inversions 148
MacMahon's Master Theorem 150
Dyson's conjecture 152
Constraints of Analytic Functions 155
References 160
Henrici's Friendly Monster Identity Revisited 162
Introduction 162
Egorychev's Method in Action 163
Reduction to a single sum 164
Simplifying the single sum 165
MultiSum in Action 166
GeneratingFunctions in Action 167
Conclusion 168
References 169
The Automatic Central Limit Theorems Generator (and Much More!) 171
References 180