Asymptotic results for the number of paths in a grid

Authors

  • Alois Panholzer
  • Helmut Prodinger

Keywords:

restricted lattice paths, asymptotic enumeration, diagonalization method, saddle point method

Abstract

In two recent papers, Albrecht and White, and Hirschhorn, respectively, considered the problem of counting the total number $P_{m,n}$ of certain restricted lattice paths in an $m \times n$ grid of cells, which appeared in the context of counting train paths through a rail network. Here we give a precise study of the asymptotic behaviour of these numbers for the square grid, extending the results of Hirschhorn, and furthermore provide an asymptotic equivalent of these numbers for a rectangular grid with a constant proportion $\alpha = m/n$ between the side lengths. DOI: 10.1017/S0004972711002759

Published

2012-04-22

Issue

Vol. 85 No. 3 (2012)

Section

Articles
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