Combinatorial Number Theory (eBook)

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BruceM. Landman, University of West Georgia, Carrollton, USA; Melvyn B. Nathanson, The City University of New York, Bronx, USA; Jaroslav Nešetril, Charles University, Prague, Czech Republic; Richard J. Nowakowski, Dalhousie University, Halifax, Canada; Carl Pomerance, Dartmouth College, Hanover,;Aaron Robertson, Colgate University, Hamilton, USA.

Preface 5
1 The Misère Monoid of One-Handed Alternating Games 11
1.1 Introduction 11
1.1.1 Background 12
1.2 Equivalences 14
1.3 Outcomes 20
1.4 The Misère Monoid 22
2 Images of C-Sets and Related Large Sets under Nonhomogeneous Spectra 25
2.1 Introduction 25
2.2 The Various Notions of Size 29
2.3 The Functions fa and ha 35
2.4 Preservation of J -Sets, C-Sets, and C*-Sets 37
2.5 Preservation of Ideals 43
3 On the Differences Between Consecutive Prime Numbers, I 47
3.1 Introduction and Statement of Results 47
3.2 The Hardy–Littlewood Prime k-Tuple Conjectures 48
3.3 Inclusion–Exclusion for Consecutive Prime Numbers 49
3.4 Proof of the Theorem 52
4 On Sets of Integers Which Are Both Sum-Free and Product-Free 55
4.1 Introduction 55
4.2 The Upper Density 57
4.3 An Upper Bound for the Density in Z/nZ 60
4.4 Examples With Large Density 61
5 Four Perspectives on Secondary Terms in the Davenport–Heilbronn Theorems 65
5.1 Introduction 65
5.2 Counting Fields in General 66
5.2.1 Counting Torsion Elements in Class Groups 69
5.3 Davenport–Heilbronn, Delone–Faddeev, and the Main Terms 70
5.3.1 TheWork of Belabas, Bhargava, and Pomerance 71
5.4 The Four Approaches 72
5.5 The Shintani Zeta-Function Approach 73
5.5.1 Nonequidistribution in Arithmetic Progressions 76
5.6 A Refined Geometric Approach 77
5.6.1 Origin of the Secondary Term 78
5.6.2 A Correspondence for Cubic Forms 79
5.7 Equidistribution of Heegner Points 80
5.7.1 Heegner Points and Equidistribution 81
5.8 Hirzebruch Surfaces and the Maroni Invariant 83
5.9 Conclusion 84
6 Spotted Tilings and n-Color Compositions 89
6.1 Background 89
6.2 n-Color Composition Enumerations 91
6.3 Conjugable n-Color Compositions 96
7 A Class ofWythoff-Like Games 101
7.1 Introduction 101
7.2 Constant Function 103
7.2.1 A Numeration System 104
7.2.2 Strategy Tractability and Structure of the P-Positions 108
7.3 Superadditive Functions 109
7.4 Polynomial 113
7.5 Further Work 116
8 On the Multiplicative Order of FnC1=Fn Modulo Fm 119
8.1 Introduction 119
8.2 Preliminary Results 120
8.3 Proof of Theorem 8.1 124
8.4 Comments and Numerical Results 130
9 Outcomes of Partizan Euclid 133
9.1 Introduction 133
9.2 Game Tree Structure 135
9.3 Reducing the Signature 138
9.3.1 Algorithm 142
9.4 Outcome Observations 143
9.5 Open Questions 144
10 Lecture Hall Partitions and theWreath Products Ck . Sn 147
10.1 Introduction 147
10.2 Lecture Hall Partitions 148
10.3 Statistics on Ck . Sn 149
10.4 Statistics on s-Inversion Sequences 150
10.5 From Statistics on Ck o Sn to Statistics on In,k 151
10.6 Lecture Hall Polytopes and s-Inversion Sequences 153
10.7 Lecture Hall Partitions and the Inversion Sequences In,k 155
10.8 A Lecture Hall Statistic on Ck . Sn 158
10.9 Inflated Eulerian Polynomials for Ck . Sn 160
10.10 Concluding Remarks 163