Essays in Combinatorics: Crystals on Shifted Primed Tableaux, Bigraded Fibonacci Numbers and Data Mining on Social Graphs

Abstract
Acknowledgments
Chapter 1. Introduction
	1.1. Crystals and type C Stanley Symmetric Functions.
	1.2. Cores with distinct parts and bigraded Fibonacci numbers.
	1.3. Estimating graphlet statistics via Lifting
Chapter 2. Crystal Analysis of type C Stanley Symmetric Functions
	2.1. Crystal isomorphism
	2.2. Explicit crystal operators on shifted primed tableaux
	2.3. Implementation in Sage
	2.4. Proof of Theorem 2.19
	2.5. Proof of Theorem 2.21
	2.6. Outlook
Chapter 3. Cores with distinct parts and bigraded Fibonacci numbers
	3.1. Simultaneous cores with distinct parts.
	3.2. Maximum size of (a,a+1)-cores with distinct parts.
	3.3. Number of (2k-1,2k+1)-cores with distinct parts.
	3.4. Graded Fibonacci numbers and (a,as+1)-cores with distinct parts.
	3.5. Bounce statistic and bigraded Fibonacci numbers.
Chapter 4. Estimating Graphlet Statistics via Lifting
	4.1. Prior sampling schemes
	4.2. Subgraph lifting
	4.3. Theoretical Analysis of Lifting
	4.4. Experiments
	4.5. Supplement to "Estimating Graphlets via Lifting"
	4.6. Conclusion
Bibliography