Lattice Path Enumeration and Analytic Combinatorics in Several Variables

This talk focusses on the interaction between the kernel method, a powerful collection of techniques used extensively in the enumeration of lattice walks in restricted regions, and the relatively new field of analytic combinatorics in several variables (ACSV). In particular, the kernel method often allows one to write the generating function for the number of lattice walks restricted to certain regions as the diagonal of an explicit multivariate rational function, which can then be analyzed using the methods of ACSV. This pairing is powerful and flexible, allowing for results which can be generalized to high (or even arbitrary) dimensions, weighted step sets, and the enumeration of walks returning to certain boundary regions of the domains under consideration. Several problems will be discussed, including joint work with Alin Bostan, Mireille Bousquet-Mélou, Julien Courtiel, Marni Mishna, Kilian Raschel, and Mark Wilson.Non UBCUnreviewedAuthor affiliation: University of Waterloo & ENS LyonGraduat