Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics

By Bryan Ek


First Written: March 29th, 2018; This version: April 16th, 2018.
Submitted to Rutgers University to fulfill the dissertation requirement of a doctoral degree.

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.

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Dissertation Components:

  • Unimodal Polynomials
  • Lattice Walk Enumeration
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