Lattice Path Counter Calculator

About this calculator

A monotonic lattice path on an m × n grid is a route from one corner to the opposite corner that only moves right or up (never left or down) along the grid lines. The number of such paths is a classic combinatorics result: since any path consists of exactly m rightward moves and n upward moves in some order, the total count is the binomial coefficient C(m+n, m) — the number of ways to choose which m of the m+n total moves are “right” moves.

This is one of the first non-trivial results taught in combinatorics and discrete mathematics courses, and it connects directly to Pascal's triangle, binomial coefficients, and the reflection principle used to solve more advanced counting problems like paths that must avoid a diagonal (the basis of the Catalan numbers). Computer science students encounter the same structure in dynamic-programming grid problems, where counting lattice paths is a standard warm-up exercise before tackling weighted or obstacle-avoiding path counts.