The Relationship Between Pascal's Triangle and Random Walks

Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schr"{o}der, and Motzkin number sequences through a combination of matrix operations, quadratic functions, and inductive reasoning. Our results revealed a number of distinct patterns, some unnamed, between these number sequences and Pascal’s triangle that can be explained through generating functions, first returns, group theory, and the Riordan matrix. Various proofs and properties of these number sequences are provided, including each generating function, their respective first returns, and matrix properties. These findings lead to a deeper understanding of combinatorics and graph theory.