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Algorithmica

, Volume 75, Issue 4, pp 782–811 | Cite as

Asymptotic Lattice Path Enumeration Using Diagonals

  • Stephen Melczer
  • Marni MishnaEmail author
Article

Abstract

We consider d-dimensional lattice path models restricted to the first orthant whose defining step sets exhibit reflective symmetry across every axis. Given such a model, we provide explicit asymptotic enumerative formulas for the number of walks of a fixed length: the exponential growth is given by the number of distinct steps a model can take, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. The generating function of each model is first expressed as the diagonal of a multivariate rational function, then asymptotic expressions are derived by analyzing the singular variety of this rational function. Additionally, we show how to compute subdominant growth, reflect on the difference between rational diagonals and differential equations as data structures for D-finite functions, and show how to determine first order asymptotics for the subset of walks that start and end at the origin.

Keywords

Lattice path enumeration D-finite Diagonal Analytic combinatorics in several variables Weyl chambers 

Notes

Acknowledgments

The authors would like to thank Manuel Kauers for the construction in Proposition 2.6, and illuminating discussions on diagonals of generating functions, and the anonymous referees and editors of both this work and its previous extended abstract for their comments and suggestions. We are also grateful to Mireille Bousquet-Mélou for pointing out some key references and provoking important clarifications.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.U. Lyon, CNRS, ENS de Lyon, INRIAUCBL, Laboratoire LIPLyonFrance
  3. 3.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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