A Cornucopia Of Labeled Diagrams And Their Generating Polynomials
Combinatorics on tableaux-like objects and understanding the relationships of various polynomial bases with each other are classical explorations in algebraic combinatorics. This type of exploration is the focus of this dissertation. In the world of symmetric polynomials and their corresponding objects, we prove some partial results for the Schur expansion of Jack polynomials in certain binomial coefficient bases. As a result, we conjecture a bijection between tableaux and rook boards, which spurs some further exploration of quasi-Yamanouchi tableaux as combinatorial objects of their own merit.
We then move to the general polynomial ring and two of its bases, key and lock polynomials. These are each generating polynomials of certain kinds of Kohnert diagrams, and we use this connection to say something about their relationship. Each of the objects that they are generating polynomials of have a nice crystal structure. We prove that the crystal structure corresponding to lock polynomials is connected and can be embedded into the crystal structure corresponding to key polynomials.