Asymptotic results for the number of paths in a grid
In two recent papers, Albrecht and White ['Counting paths in a grid', Austral. Math. Soc. Gaz. 35 (2008), 43-48] and Hirschhorn ['Comment on "Counting paths in a grid", Austral. Math. Soc. Gaz. 36 (2009), 50-52] considered the problem of counting the total number Pm,n of certain restricted lattice paths in an m × 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 α = m/n between the side lengths. © 2011 Australian Mathematical Publishing Association Inc.