Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics

TL;DR

This dissertation develops flexible, algorithmic methods to analyze unimodality of polynomials and lattice walk enumeration, enabling the discovery of new results and generalizations in combinatorics.

Contribution

It simplifies and extends existing methods for proving unimodality and lattice walk enumeration, making them more adaptable and automatable for complex cases.

Findings

Created new families of unimodal polynomials.

Developed an algorithmic approach for lattice walk enumeration.

Discovered new results for complex step sets.

Abstract

The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. Ayyer and Zeilberger proved a result about bounded lattice walks. I employ their generating function relation technique to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded planes. The method in which we do this is formulated to be highly algorithmic so that a computer can automate most, if not all, of the work. I easily recover many well-known results for simpler step sets and discover new results for more complex step sets.