Algorithmic Approaches for Lattice Path Combinatorics

Abstract

Lattice paths are fundamental combinatorial classes and they appear in many guises in physics, queuing theory, language theory, and pure mathematics. New formulas for the asymptotic enumeration of walks restricted to a quadrant have appeared in the past ten years, mostly driven by novel systematic, and analytic approaches. In this tutorial we will present the universe of lattice path classes, and survey some of the strategies used to deduce enumerative formulas. The generating functions for several models of walks in cones satisfy nice differential equations, and algorithms to find, manipulate, and solve differential equations have been central to this study. We will discuss how to express the generating functions as diagonals of multivariate rational functions, and how to reconcile the results of different approaches to this computation.