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Probability and Combinatorics

Tuesday, October 3, 2017 - 3:00pm

Stephen Melczer

University of Pennsylvania


University of Pennsylvania


The problem of enumerating lattice paths with a fixed set of allowable steps and restricted endpoint has a long history dating back at least to the 19th century. For several reasons, much research on this topic over the last decade has focused on two dimensional lattice walks restricted to the first quadrant, whose allowable steps are "small" (that is, each step has coordinates +/- 1, or 0). In this talk we relax some of these conditions and discuss recent work on walks in higher dimensions, with non-small steps, or with weighted steps. Particular attention will be given to the asymptotic enumeration of such walks using representations of the generating functions as diagonals of rational functions, through the theory of analytic combinatorics in several variables.  Several techniques from computational and experimental mathematics will be highlighted, and open conjectures of a probabilistic nature will be discussed.