TOPICS
Lattice Path
A lattice path is a sequence of points 
, 
,
..., 
with 
such that each 
is a lattice point and
each difference 
belongs to a specified set of allowed steps. For unit horizontal
and vertical steps, 
is obtained by offsetting one unit east (or west) or one unit north (or south). Other
step sets may include diagonal steps (Rohatgi 1964, p. 470).
The number of paths of length 
from the origin (0,0) to a point
which are restricted to east and
north steps is given by the binomial coefficient 
for nonnegative
integers 
and 
.
See also
Ballot Problem, Delannoy Number, Delannoy Polynomial, Dyck Path, Faber Polynomial, Golygon, Kings Problem, Lattice Point, p-Good Path, Random Walk, Staircase WalkExplore with Wolfram|Alpha
References
Dickau, R. M. "Shortest-Path Diagrams." https://www.robertdickau.com/manhattan.html.Hilton, P. and Pederson, J. "Catalan Numbers, Their Generalization, and Their Uses." Math. Intel. 13, 64-75, 1991.Mohanty, S. G. Lattice Path Counting and Applications. New York: Academic Press, 1979.Moser, L. and Zayachkowski, W. "Lattice Paths with Diagonal Steps." Scripta Math. 26, 223-229, 1963.Narayana, T. V. Lattice Path Combinatorics with Statistical Applications. Toronto, Ontario, Canada: University of Toronto Press, 1979.Rohatgi, V. K. "On Lattice Paths with Diagonal Steps." Canad. Math. Bull. 7, 470-472, 1964. https://doi.org/10.4153/CMB-1964-046-8.Referenced on Wolfram|Alpha
Lattice PathCite this as:
Weisstein, Eric W. "Lattice Path." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LatticePath.html