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ASYMPTOTIC RESULTS FOR THE NUMBER OF PATHS IN A GRID

Part of: Combinatorics

Published online by Cambridge University Press:  04 October 2011

ALOIS PANHOLZER  and
HELMUT PRODINGER
ALOIS PANHOLZER
Affiliation:
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Haupstr. 8-10/104, 1040 Wien, Austria (email: Alois.Panholzer@tuwien.ac.at)
HELMUT PRODINGER*
Affiliation:
Mathematics Department, Stellenbosch University, 7602 Stellenbosch, South Africa (email: hproding@sun.ac.za)
*
✉For correspondence; e-mail: hproding@sun.ac.za
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Abstract

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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.

Keywords

restricted lattice pathsasymptotic enumerationdiagonalization methodsaddle point method

MSC classification

Secondary: 05A16: Asymptotic enumeration
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 3 , June 2012 , pp. 446 - 455
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Abramovitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 (US Government Printing Office, Washington, DC, 1964).CrossRefGoogle Scholar
[2]Albrecht, A. R. and White, K., ‘Counting paths in a grid’, Austral. Math. Soc. Gaz. 35 (2008), 43–48.Google Scholar
[3]Flajolet, P. and Sedgewick, R., Analytic Combinatorics (Cambridge University Press, Cambridge, 2009).CrossRefGoogle Scholar
[4]Hautus, M. L. J. and Klarner, D. A., ‘The diagonal of a double power series’, Duke Math. J. 38 (1971), 229–235.CrossRefGoogle Scholar
[5]Hirschhorn, M. D., ‘How many ways can a king cross the board?’, Austral. Math. Soc. Gaz. 27 (2000), 104–106.Google Scholar
[6]Hirschhorn, M. D., ‘Comment on “Counting paths in a grid”’, Austral. Math. Soc. Gaz. 36 (2009), 50–52.Google Scholar