### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu
**Bryan Ek**, Rutgers University, bryan [dot] t [dot] ek {at} math [dot] rutgers [dot] edu

Title: Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics (Thesis Defense)

Speaker: **Bryan Ek**, Rutgers University

Date: Thursday, March 29, 2018 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The main goal of these projects was utilizing experimental mathematics to further our knowledge of several areas in math.
We begin by tweaking a proof of unimodality by O'Hara to produce many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. We then employ a generating function relation technique used by Ayyer and Zeilberger 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.

See: http://sites.math.rutgers.edu/~bte14/expmath/