UC Davis

The Catalan numbers are a ubiquitous sequence of natural numbers appearing in a diversearray of mathematical fields. However, even though these numbers have been well-studied, several conjectures and properties surrounding the Catalan numbers remain open. In this dissertation we first study the joint distribution of various statistics defined on Dyck paths. The first joint distribution involves the area and diagonal inversion statistic in the form of the q, t-Catalan polynomial. This polynomial arises from the study of the space of diagonal harmonics, and its symmetry has evaded a combinatorial proof. We introduce two new q, t-Catalan polynomials using two new statistics on Dyck paths. We are able to give a combinatorial proof of their symmetry and recover the usual q, t-Catalan polynomial in terms of our new statistics. Next, we explore the joint distribution of NE and NNE-factors within Dyck paths. We answer an open question by Bóna and Labelle regarding the symmetry of these numbers at certain values. Additionally, we prove various enumerative results of these numbers, including their real-rootedness and their connection to the number of cyclic compositions.

Kashiwara’s crystal bases are combinatorial structures introduced in his study of the representations of quantum groups under a certain limit. Using Kashiwara’s crystals, we explore the Burgecorrespondence sending labelled graphs to tableau. We give a Schensted-like result characterizing when a labelled graph is sent to a hook-shaped tableau and give a type A crystal structure on such graphs. Lastly, we merge these two topics by looking at the space of invariant tensors of the spin and vector representations in Type B. Using the promotion operator on Kashiwara’s crystals, we construct a diagrammatic basis for these spaces in terms of chord diagrams such that rotation of the chord diagrams intertwines with the cyclic action on tensor factors. As a consequence of this, we are able to give a cyclic sieving phenomenon for fans of Dyck paths and vacillating tableaux respectively.