Tool to calculate all paths on a lattice graphe (square grid graph). A path is a series of directions (north, south, east, west) to connect two points on a grid.

Lattice Path - dCode

Tag(s) : Graph Theory

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The information on this page is for a square grid and is not valid on triangular grids (or other non square lattice graphs).

The calculation of the number of paths (of length $ a + b $) on a grid of size (a x b) (limited to a north-south direction and a west-east direction) uses combinatorics tools such as the binomial coefficient $ \binom{a+b}{a} $

The north direction N consists of moving up one unit along the ordinate (0,1).

The east direction E consists of moving one unit to the right along the abscissa (1,0).

__Example:__ To go from the point $ (0, 0) $ to the point $ (2, 2) $ (which corresponds to a 2x2 grid) using only north and east. `(N,N,E,E)`, `(N,E,N,E)`, `(N,E,E,N)`, `(E,N,E,N)`, `(E,N,N,E)`, `(E,E,N,N)` so 6 paths and is computed $ \binom{4}{2} = 6 $

A grid graph is the name given to a bounded grid (with borders).

To generate the list of all paths, use the permutation generator.

__Example:__ `N,N,N,E` has 4 distinct permutations: `(N,N,N,E) (N,N,E,N) (E,N,N,N) (N,E,N,N)`

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Cite as source (bibliography):

*Lattice Path* on dCode.fr [online website], retrieved on 2024-11-11,

path,lattice,grid,combinatoric,point

https://www.dcode.fr/lattice-path

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