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

Author Bio

Shakuan Frankson is in the second year of her Mathematics PhD program at Howard University. She received a fellowship and several scholarships from her home institution, presented at several conferences and events, such as NAM MathFest and MAA Undergraduate Programming and Diversity Focus Day, participated in the Honda Campus All-Star Challenge (HCASC) club, and tutored undergraduate students in various levels of mathematics. She is also a proud member of Pi Mu Epsilon, the honorary national mathematics society, and Sigma Alpha Pi, the National Society of Leadership and Success.

Myka Terry is a senior Mathematics major, Physics minor at Morgan State University. She is a LSAMP recipient and student researcher, the president of MSU's Math Club, Miss Afros of MSU, the president of Morgan State's Live Squad, and a campus tour guide. She has also presented at several conferences, such as ABRCMS and CAARMS24 hosted by Princeton University. Myka is also interested in pursuing further research in discrete mathematics and combinatorics.