Combinatorics of Set Partitions

Introduction Historical Overview and Earliest Results Timeline of Research for Set Partitions A More Detailed Book Basic Tools of the Book Sequences Solving Recurrence Relations Generating Functions Lagrange Inversion Formula The Principle of Inclusion and Exclusion Generating Trees Preliminary Results on Set Partitions Dobinski's Formula Different Representations Subword Statistics on Set Partitions Subword Patterns of Size Two: Rises, Levels and Descents Peaks and Valleys Subword Patterns: l-Rises, l-Levels, and l-Descents Families of Subword Patterns Patterns of Size Three Nonsubword Statistics on Set Partitions Statistics and Block Representation Statistics and Canonical and Rook Representations Records and Weak Records Number of Positions between Adjacent Occurrences of a Letter The Internal Statistic Statistics and Generalized Patterns Major Index Number of Crossings, Nestings and Alignments Avoidance of Patterns in Set Partitions History and Connections Avoidance of Subsequence Patterns Generalized Patterns Partially Ordered Patterns Multi Restrictions on Set Partitions Avoiding a Pattern of Size Three and Another Pattern Pattern Avoidance in Noncrossing Set Partitions General Equivalences Two Patterns of Size Four Left Motzkin Numbers Sequence A054391 Catalan and Generalized Catalan Numbers Pell Numbers Regular Set Partitions Distance Restrictions Singletons Block-Connected Asymptotics and Random Set Partition Tools from Probability Theory Tools from Complex Analysis Z-Statistics Set Partitions as Geometric Words Asymptotics for Set Partitions Gray Codes, Loopless Algorithms and Set Partitions Gray Code and Loopless Algorithms Gray Codes for Pn Loopless Algorithm for Generating Pn Set Partitions and Normal Ordering Preliminaries Linear Representation and N((a+a)n) Wick's Theorem and q-Normal Ordering p-Normal Ordering Noncrossing Normal Ordering Appendices Bibliography Index Exercises, Research Directions, and Open Problems appear at the end of each chapter.