Hopf Algebras of Quasi-Symmefric Functions

Notation viii
1 Introduction 1
1.1 Why Study Quasi-Symmetric Functions? ................................... 1
1.2 Compositions .................................................................................. 7
1.3 Partially Ordered S e ts .................................................................. 10
1.4 Hopf Algebras .................................................................................. 12
1.5 The Hopf Algebra of Symmetric Functions ............................... 19
2 The Hopf Algebra of Quasi-Symmetric Functions 28
2.1 Quasi-Symmetric Functions ......................................................... 28
2.2 Ordinary P-Partitions .................................................................. 35
2.3 The Hopf Algebra of Noncommutative Symmetric Functions . 47
2.4 Combinatorial Hopf A lgebras...................................................... 48
2.5 The Hopf Algebra of Perm utations ............................................ 50
3 The Hopf Algebra of Peak Quasi-Symmetric Functions 54
3.1 Peak Quasi-Symmetric Functions ............................................... 54
3.2 Enriched P-Partitions .................................................................. 60
3.3 Eulerian Combinatorial Hopf Algebras 71
4 The Hopf Algebra of Shifted Quasi-Symmetric Functions 75
4.1 Shifted Quasi-Symmetric Functions ............................................. 75
4.2 General Enriched P -P artitions .................................................... 78
Future Work 110
Bibliography 113