Random Walk Combinatorial Identities

We explore visually two different types of random walks in two dimensions. In a traditional random walk, a single decision is made at each step to go up, down, left, or right. In a diagonal random walk, two decisions are made at each step, to go up or down, and to go left or right, resulting in diagonal steps. We count the number of paths to landing points in each type. In a traditional random walk, the path counting is done using a sum of multinomial coefficients. In a diagonal random walk, we use a product of binomial coefficients. We notice an isomorphism between the two problems, leading to interesting and elegant combinatorial identities, some already known and some possibly new.