# [seqfan] Re: lattice paths

Gerald McGarvey Gerald.McGarvey at comcast.net
Thu Oct 1 04:10:39 CEST 2009

```Brendan,

For the vertical step set {-3,2} case, the number of ways to go to (5*n,0)
seems to be A060941(n). These values are also in the (x,3) row.
Also, for the {-2,3} case these values are in (5*n,0) and the (x,3)
and (x,5) rows.

<http://www.research.att.com/%7Enjas/sequences/A060941>A060941
Duchon's numbers: the number of paths of length 5*n from the origin
to the line y=2*x/3 with unit East and North steps that stay below
the line or touch it.
1, 2, 23, 377, 7229, 151491, 3361598, 77635093, 1846620581,
44930294909, 1113015378438, 27976770344941, 711771461238122,
18293652115906958, 474274581883631615, 12388371266483017545,
325714829431573496525

http://www.labri.fr/perso/duchon/
http://www.labri.fr/perso/duchon/index-en.html

Best regards,
Gerald

At 05:38 AM 8/10/2009, Brendan McKay wrote:
>Consider walks in { (x,y) | x,y >= 0 } with steps of
>size (1,2) and (1,-3), starting at (0,0).
>
>Typical walk: (0,0), (1,2), (2,4), (3,1), (4,3), (5,0), (6,2).
>
>This example has vertical step set {-3,2} but of course it
>could be any set of integers.
>
>Where are such walks analysed? Surely there is a definitive
>treatment somewhere, but the literature on lattice walks is so vast
>that finding the right trees amongst all the wood is a bit of a
>struggle.
>
>Thanks,
>Brendan.
>
>
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>
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```