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A probabilistic method for lattice path enumeration. (English) Zbl 0602.05006
This paper presents a probabilistic method for obtaining functional equations for lattice path enumeration problems. A number of problems are handled in this manner (about half of these are listed below); for most of them, a combinatorial solution is either offered or cited, but usually the probabilistic solution involves less computation.
1. Given positive integers r and s, find $$d_ n$$, the number of paths in the plane with unit steps up or right from (0,0) to $$(n,r+sn)$$ which never touch the line $$y=r+sx$$ except at the endpoint (solved combinatorially in [S. G. Mohanty, Lattice path counting and applications (1979; Zbl 0455.60013), p. 9]).
2. Given the boundary lines $$y=x+a$$ and $$y=x-b$$ for positive integers a, b, find $$f_ n$$, the number of paths from (0,0) to $$(m,a+m)$$ which never touch any boundary points except at the end, and $$g_ n$$, the number of analogous paths ending at $$(b+n,n).$$
3. Problem 1 except that r and s are arbitrary non-negative rationals, and the path can neither touch or cross the boundary except at the end. The case where r and s are half-integers is treated here, and a special subcase was solved combinatorially in [ibid., p. 10].
4. Find $$a_{m,n}$$, the number of paths with steps (1,0,0), (0,1,0) and (0,0,1) from (0,0,1) to $$(m,m+n+1,m+n+1)$$ which never touch the planes $$x=z$$ or $$y=z$$ except at the end (solved combinatorially in [G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du B.U.R.O. 6, 5-105 (1965)]).
The probabilistic method (which involves computing in two ways the probability that a random walk will touch the boundary somewhere) is illustrated for the first problem only; thenceforth the functional equations obtained by this method are stated immediately and then solved.
Reviewer: T.Walsh

##### MSC:
 05A15 Exact enumeration problems, generating functions 60G50 Sums of independent random variables; random walks 05C30 Enumeration in graph theory 05C38 Paths and cycles
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##### References:
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