2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 401406
author:  Philippe Nadeau 

title:  Walks reaching a line 
keywords:  walk, generating function, rational language, singularity analysis 
abstract: 
We enumerate walks in the plane
ℝ
, with steps East and North, that stop as soon as
they reach a given line; these walks are counted according
to the distance of the line to the origin, and we study the
asymptotic behavior when the line has a fixed slope and
moves away from the origin. When the line has a rational
slope, we study a more general class of walks, and give
exact as well as asymptotic enumerative results; for this,
we define a nice bijection from our walks to words of a
rational language. For a general slope, asymptotic results
are obtained; in this case, the method employed leads us to
find asymptotic results for a wider class of walks in
2
ℝ
.
m

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reference:  Philippe Nadeau (2005), Walks reaching a line, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 401406 
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