2018/04/16 by Bryan Ek, Ek, Bryan
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #DNA and Biological Computing #Data Management and Algorithms #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1804.05933
openalex publication_date 2018/04/16 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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.