Multivariate Diagonals, D-finite Functions, and Lattice Path Enumeration
| Speaker: | Stephen Melczer |
|---|---|
| Affiliation: | University of Waterloo |
| Room: | Mathematics and Computer Building (MC) 5168 |
Abstract:
In this talk we look at the efficacy of encoding D-finite generating functions (those satisfying linear differential equations) as diagonals of multivariate rational functions, with a particular focus on lattice path models in restricted regions. By combining the popular ?kernel method? for lattice path problems with recent results in the field of analytic combinatorics in several variables, this approach allows us to determine general formulas for the dominant asymptotics of counting sequences of certain symmetric models restricted to d-dimensional orthants. The exponential growth of each model is given by the number of steps, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. These expressions are derived by analyzing the singular variety of a multivariate rational function whose diagonal counts the lattice paths in question.