## Dissertations, Master's Theses and Master's Reports

2021

#### Document Type

Open Access Master's Thesis

#### Degree Name

Master of Science in Mathematical Sciences (MS)

Department of Mathematical Sciences

William J. Keith

David Hemmer

Melissa Keranen

Fabrizio Zanello

#### Abstract

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 parts. In Chapter 3 we aim to reduce the complexity of Kirillov and Reshetikhin's formula by identifying the summands contributing a nonzero amount to the polynomial. While the resulting formulas are lengthy, they greatly reduce the computation time for specified partition shapes and numbers of descents. In Chapter 4 we investigate an apparent relation among $f_{\lambda, k}(q)$ and $f_{\lambda, k-1}(q)$ and discuss how this may lead to a greater insight of the distribution of these statistics. Included appendices give a library of utilities in SageMath and Mathematica to generate the polynomials $f_{\lambda, k}$ and demonstrate Chapter 4's relationships.