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Lattice Path

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.

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Lattice Path -

Tag(s) : Graph Theory

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Lattice Path

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Path Count Calculator (North-East - NE)

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

Whole Lattice Graph



Between 2 points





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.

Answers to Questions

How to count paths on a lattice graph?

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 \( \binom{4}{2} = 6 \)

What is a lattice graph?

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

How to enumerate pathways in a lattice graph?

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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