"William Elliot" wrote in news:Pine.NEB.email@example.com... IV wrote: >>> 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?
IV wrote: >> Please concentrate your mind on this: I assume, the problem of given >> numbers of steps of each kind is a new quality of lattice path problems. >> Has someone experiences, literature references or ideas for this kind of >> problems?
> Who's to know what your talking about when you don't define your terms?
I mean only lattice paths, not lattice walks.
Fray, R. D.; Roselle, D. P.: Weighted lattice paths. Pacific J. Math. 37 (1971) (1) 85-96: "The number, binomial(m+n,n), of unrestricted minimal lattice paths from (0,0) to (m,n) is an old and well-known result, see MacMahon (MacMahon, P. A.: Combinatory Analysis, 1915/1916) or Feller (Feller, W.: An Introduction to Probability Theory and Its Applications, 1968)."
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 x-direction and one move of length s in the y-direction, s a nonnegative integer.
The lattice paths with n1 steps (1,1), n2 steps (1,2), n3 steps (1,3), ... and 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?