Lattice Path Counter Calculator

How it is calculated

A monotonic path from one corner of an m × n grid to the opposite corner takes exactly m steps right and n steps up, in any order. Choosing which m of the m + n steps go right gives paths = C(m+n, m) = (m+n)! ÷ (m! · n!). Here m and n are the number of unit moves (grid cells) along each side, not the number of grid points. The tool multiplies and divides in whole numbers using the smaller of m and n, so the count is exact with every digit shown (250 × 250 gives a 150-digit number), and it also reports the total steps (m + n) and the grid size. Inputs must be non-negative whole numbers and m + n may not exceed 500; decimals, negatives or larger grids give no result.

Common mistakes

• Entering the number of grid points (intersections) instead of cells. A grid of 5 × 4 points is a 4 × 3 grid of moves, so enter 4 and 3.
• Typing m + n above 500 or a decimal such as 2.5; the tool shows no result for either rather than approximating.
• Expecting paths that may go left or down. The count covers only right and up moves, so routes that backtrack are not included.
• Using it for grids with blocked cells or a forbidden diagonal; the plain C(m+n, m) formula ignores obstacles.
• Assuming a zero side means no path. With m = 0 or n = 0 there is exactly one straight path.