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arXiv · 1804.05933

Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics

Bryan Ek
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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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Bryan Ek. 2018-04-16. Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics. https://arxiv.org/abs/1804.05933

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