By definition, a first return is the immediate moment that a path, using vectors in the Cartesian plane, touches the x-axis after leaving it previously from a given point; the initial point is often the origin. In this case, using certain diagonal and horizontal vectors while restricting the movements to the first quadrant will cause almost every first return to end at the point (2n,0), where 2n counts the equal number of up and down steps in a path. The exception will be explained further in the sections below. Using the first returns of Catalan, Schröder, and Motzkin numbers, which resulted from the lattice paths formed using a combination of diagonal and/or horizontal vectors, we then investigated the effect that coloring select vectors will have on each of the original generating
Leon Woodson (email@example.com)
Frankson, Shakuan and Terry, Myka
"Investigating First Returns: The Effect of Multicolored Vectors,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 21
, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol21/iss1/5