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Constructing all NE-lattice paths from $(0,0)$ to $(m,n)$

asked 2019-04-11 01:50:27 -0500

joakim_uhlin gravatar image

updated 2019-04-11 01:51:20 -0500

If I consider only Dyck Paths, I can do write something like this:

DWS=DyckWords(3).list()
for D in DWS:
    print(D.height())

to obtain the height of all Dyck paths of length $3$. However, I would like to do the same thing but using $NE$-lattice paths from $(0,0)$ to $(m,n)$. Is there an easy way to do this in Sage?

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You could try using a bijection with something that is in Sage, e.g. see Ferrers diagram of a partition.

rburing gravatar imagerburing ( 2019-04-11 02:43:27 -0500 )edit

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answered 2019-04-11 08:33:37 -0500

joakim_uhlin gravatar image

I fixed this myself but in case anyone is interested I will leave this as an answer.

I decided to use the kbits from this thread to generate all binary strings of length $n$ with $m$ ones. Such strings are in a natural bijection with $NE$-lattice paths from $(0,0)$ to $(n,m)$. After that, I can get the heights of all the paths by writing

Paths = WordPaths('01', steps=[(1,-1),(1,1)])

for b in kbits(m+n,m):
    print(Paths(b).ymax())

Note: Sage has a "height" function on words, but this is not what I am looking for. In the definition of height that I am using, the height of an $NE$-path is geometrically the biggest distance that the path is away from the main diagonal.

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Asked: 2019-04-11 01:50:27 -0500

Seen: 44 times

Last updated: Apr 11