Search for a tool
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.

Results

Lattice Path -

Tag(s) : Graph Theory

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Lattice Path' tool for free! Thank you!

# Lattice Path

## 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).

### 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)

## Source code

dCode retains ownership of the "Lattice Path" source code. Except explicit open source licence (indicated Creative Commons / free), the "Lattice Path" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Lattice Path" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Lattice Path" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Lattice Path" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Lattice Path on dCode.fr [online website], retrieved on 2024-06-15, https://www.dcode.fr/lattice-path

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!