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Staircase Walk


StaircaseWalk

The number of staircase walks on a grid with m horizontal lines and n vertical lines is given by

 (m+n; m)=((m+n)!)/(m!n!)

(Vilenkin 1971, Mohanty 1979, Narayana 1979, Finch 2003). The first few values for m=n=1, 2, ..., are 1, 2, 6, 20, 70, 252, ... (OEIS A000984), which are the central binomial coefficients. A Dyck path is a staircase walk from (0,0) to (n,n) which never crosses (but may touch) the diagonal y=x.


See also

Catalan Number, Central Binomial Coefficient, Dyck Path, Ferrers Diagram, Staircase Polygon

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References

Finch, S. R. "Self-Avoiding Walk Constants." §5.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 331-339, 2003.Mohanty, S. G. Lattice Path Counting and Applications. New York: Academic Press, 1979.Narayana, T. V. Lattice Path Combinatorics with Statistical Applications. Toronto, Ontario, Canada: University of Toronto Press, 1979.Sloane, N. J. A. Sequence A000984/M1645 in "The On-Line Encyclopedia of Integer Sequences."Vilenkin, N. Ya. Combinatorics. New York: Academic Press, 1971.

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Staircase Walk

Cite this as:

Weisstein, Eric W. "Staircase Walk." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StaircaseWalk.html

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