A lattice path is a path in a grid or lattice that consists of a sequence of steps between points in the grid. Typically, a lattice path is defined within a two-dimensional square grid, where the points are represented by pairs of non-negative integers \((x, y)\), and the path is composed of steps that move in specific directions. In the most common cases, the steps are restricted to two directions: right (R) and up (U).
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A lattice path is a sequence of points in an integer lattice whose consecutive differences belong to a prescribed finite set of steps.
For instance, a north-east lattice path uses only unit steps east or north in the integer lattice . Restricting the allowed steps makes the counting question precise: the endpoint fixes how many of each step must occur, but their order remains free. Pascal's rule counts these lattice paths by their last step; the same binomial coefficient counts selections of step positions.