
Re: Number of lattice paths with given steps?
Posted:
Nov 22, 2012 9:44 PM


On Thu, 22 Nov 2012, IV wrote: > "William Elliot" wrote in
> > > > I'm looking for a formula for calculating the number of lattice > > > > paths in a rectangular quadratic integer lattice with given kind > > > > of steps and given numbers of steps of each kind. That means all > > > > lattice paths with n1 steps (1,1), n2 steps (1,2), n3 steps (1,3) > > > > and so on  the numbers n1, n2, n3, ... are given. Was this > > > > problem already discussed in the literature? > > I mean only lattice paths, not lattice walks. > Fray, R. D.; Roselle, D. P.: Weighted lattice paths. Pacific J. Math. 37
Weighted paths don't apply.
> Ch. A. Charalambides: "Enumerative Combinatorics", 2002: "Consider an > orthogonal system of axes on the plane xy and the lattice defined by the > straight lines x=i, i=0,+1, +2,..., and y=j, j=0,+1,+2,..., that are > orthogonal to the (horizontal) axis x and the (vertical) axis y, > respectively. A directed polygonal line on the lattice that leads from the > point (r,s) to the point (n,k), r<=n, s<=k, through horizontal and vertical > straight sections, positively directed, is called lattice path (or, more > precisely, minimal lattice path) from the point (r,s) to the point (n,k)." > > A step is a line from one lattice point to an adjacent lattice point. > > My steps (1,y), better: (1,s), mean one move of length 1 in the xdirection > and one move of length s in the ydirection, s a nonnegative integer. The notation is inconsistent unto usless meaningless.
> The lattice paths with n1 steps (1,1), n2 steps (1,2), n3 steps (1,3), ... and
Is (1,1) in the x direction or the y direction? Is (1,2) in the x direction or the y direction? Is (1,3) in the x direction or the y direction?
What ever the direction, does the first 1 means a single step from one point to an adjecent point?
> nm steps (1,m)  the numbers n1, n2, n3, ...  nm are given, describe the > compositions of an integer n which belong to a given partition of the integer > n. Their number is m!/(n1!*n2!*...*nm!). If the lattice paths would be not > restricted, we can construct the binomial expression of their numbers by > summing up the numbers of a number triangle of the lattice path numbers, e.g. > Pascal's triangle. But how can we get a formula for the construction of the > single lattice paths which are restricted by my conditions above?

