Combinatorial Interpretations of Hankel Matrices and Further Combinatorial Uses of Riordan Group Methods

DISSERTATION APPROVAL SHEET ......................................................... ii
DEDICATION ........................................................................................................ iii
ACKNOWLEDGMENTS .................................................................................. iv
ABSTRACT ............................................................................................................ vi
LIST OF FIGURES .............................................................................................. x
LIST OF TABLES ................................................................................................ xii
CHAPTER
1. Introduction ..................................................................................................... 1
1.1 Hankel Matrices and Determ inants ......................................................... 1
1.2 The Riordan Group ...................................................................................... 4
1.3 Directed Graphs ................................................................................................. 10
2. Schroder Numbers and Hankel Determinants ..................................... 12
2 .1 Schroder Numbers ........................................................................................ 12
2.2 Hankel Determinants of Schroder Numbers .............................................. 15
2.3 Combinatorial Interpretations ........................................................................ 29
3. Generalization of Schroder Numbers R esu lts ....................................... 31
3.1 m -adm issible ..................................................................................................... 31
3.2 4-admissible Sequence ..................................................................................... 33
3.2.1 4,-admissible Translation ...................................................................... 35
3.2.2 Schroder Analog ................................................................................... 36
3.2.3 Free Bishop M oves ............................................................................... 37
3.2.4 Up Bishop Moves ................................................................................. 40
3.3 Bijective proofs ................................................................................................. 43
3.3.1 4-admissible Translation Schroder Analog .......................... 43
3.3.2 Free Bishop Moves <*=>• Schroder A nalog ....................................... 47
3.3.3 Up Bishop Moves Schroder Analog ....................................... 51
3.4 Main R esult ....................................................................................................... 58
3.5 Additional Combinatorial Interpretations ..................................................... 59
4. Disjoint Motzkin Path Systems .................................................................. 62
4.1 Disjoint paths and Hankel Determinants ..................................................... 62
4.2 Motzkin Paths ................................................................................................... 63
5. Conclusion ........................................................................................................... 72
5.1 Open Questions
REFERENCES .........