Anyone know much about combinatorics?

[LAHLH]

Hi,

I'm looking at the problem of counting lattice paths between two points with n steps, and I think I have my head around the case without boundaries to bump into (see e.g. http://www.robertdickau.com/lattices.html). However I'd like to work out the answer for the case of my path not being able to cross two boundaries , I know the answer is given here http://www.math.ucdavis.edu/~blakehunter/Masters.pdf , on p10 Lemma 1-1.1, and I can sort of see this going to be related to the case without boundaries and catalan numbers, but I would like to flesh out the details...

I believe the proof is given in "Mohanty, S. G., Lattice Path Counting and Applications", and also "Narayana, T.V., Lattice Path Combinatorics With Statistical Applications" but both of these seem out of print and even my uni library doesn't have them, nor can I seem to find them online. Does anyone know where I can get a copy or somewhere else that does the proof of the above? or just how to prove the above fullstop?

thanks

 

[LAHLH]

noone out there know how to do this? It's not my field but I think it's a pretty standard problem in combinatorics if I could only find the books..

I think you need to use the reflection principle a few times, but I'm not sure on the specifics. Bit like the ballot problem but with two boundaries that can't be crossed.