The main notes and results are here.
The following work originated during my summer as part of Stanford Undergraduate Research in Mathematics (SURIM). I worked in a group with Arianna Serafini and Mason Rogers under the guidance of then Stanford Math Ph.D. Canditiate Erik Bates. Our work from the summer is summarized in our final report here. From this work, our group presented two posters at the Joint Mathematics Meetings in 2018.
My contributions from the above document are largely encompassed in Section 3. Another student in SURIM, Patrick Revilla, became interested in the sorts of problems I was working on during the end of the summer. We began to work together on future avenues of research as I was writing the report. We continued work (very) intermittently on the problems after the program.
We provide two main results in the notes found here. First, we count the number of equivalence classes that are induced on binary strings of length n by a relation r. Second, we are able to come up with a complete classification theorem for relational trees. See the notes for definitions.
I'd like to thank Erik for his guidance during the summer, and Arianna and Mason for both providing an open ear when I wanted to bounce ideas off someone and for making helpful suggestions throughout the research process. I'd like to especially thank Patrick for his continued interest in the problem, and his helpful ideas surrounding developing machinery and contributions to the work.