Discrete Mathematics and Combinatorics Commons^™

Integer Partitions Under Certain Finiteness Conditions, Tim Wagner

Dissertations, Master's Theses and Master's Reports

This dissertation focuses on problems related to integer partitions under various finiteness restrictions. Much of our work involves the collection of partitions fitting inside a fixed partition $\lambda$, and the associated generating function $G_{\lambda}$.

In Chapter 2, we discuss the flawlessness of such generating functions, as proved by Pouzet using the Multicolor Theorem. We give novel applications of the Multicolor Theorem to re-prove flawlessness of pure $O$-sequences, and show original flawlessness results for other combinatorial sequences. We also present a linear-algebraic generalization of the Multicolor Theorem that may have far-reaching applications.

In Chapter 3, we extend a technique due to …

Major Index Over Descent Distributions Of Standard Young Tableaux, Emily Anible

Dissertations, Master's Theses and Master's Reports

This thesis concerns the generating functions $f_{\lambda, k}(q)$ for standard Young tableaux of shape $\lambda$ with precisely $k$ descents, aiming to find closed formulas for a general form given by Kirillov and Reshetikhin in 1988. Throughout, we approach various methods by which further closed forms could be found. In Chapter 2 we give closed formulas for tableaux of any shape and minimal number of descents, which arise as principal specializations of Schur functions. We provide formulas for tableaux with three parts and one more than minimal number of descents, and demonstrate that the technique is extendable to any number of …