Dyck paths

Dyck paths are mathematical diagrams that visualize balanced sequences of steps, typically represented on a grid. Imagine starting at a point and moving only upward or downward, never dipping below the starting level, with the total steps returning to the original height. These paths correspond to well-formed sequences of parentheses or certain combinatorial structures. They are used in combinatorics to count and analyze patterns that require balance and structure, such as correctly nested expressions or binary tree configurations. Essentially, Dyck paths provide a visual way to understand and count balanced arrangements within a structured system.