Dyck Path Counter

About this calculator

A Dyck path is a staircase-like lattice path made of "up" and "down" steps that starts and ends at the same height and never dips below its starting level — equivalent to a correctly balanced sequence of parentheses, where an "up" step is an opening bracket and a "down" step is a closing one. This calculator counts how many distinct Dyck paths exist for a given semilength n (n up-steps and n down-steps, 2n steps total), a count that equals the nth Catalan number, C(2n,n)/(n+1).

Catalan numbers via Dyck paths show up constantly in combinatorics and computer science: they count the ways to correctly match parentheses in an expression, the number of distinct binary search trees with n nodes, the ways to triangulate a convex polygon, and valid push/pop sequences on a stack — which is why the same C(2n,n)/(n+1) formula keeps reappearing across seemingly unrelated counting problems in discrete math courses.