Catalan Number Sequences and Generalized Action Graphs

Authors

  • Drew Caldwell Indiana University Kokomo
  • Ali Cochran Indiana University Kokomo
  • Nathan Glisson Marian University
  • Bryce Jennings Indiana University Kokomo
  • Katy McDicken Marian University
  • Luke James Proctor Marian University
  • Sarah Klanderman Marian University
  • Amelia Tebbe Indiana University Kokomo

DOI:

https://doi.org/10.33043/r2y588ab

Abstract

Action graphs emerged from work of Bergner and Hackney on category actions in the context of Reedy categories. Alvarez, Bergner, and Lopez showed that action graphs could be inductively generated without reference to category actions, and they proved that the number of vertices added to An is the n-th Catalan number.

Author Biographies

Drew Caldwell, Indiana University Kokomo

Drew Caldwell is an undergraduate student studying Mathematics and Computer Science at Indiana University Kokomo who is expected to graduate in May of 2025. Drew graduated from Western High School in Russiaville, Indiana. Drew chose to study mathematics after teaching himself how to learn math in preparation for the SAT. His favorite theorem is the "Shoes and Socks theorem", and favorite math class is Calculus II.

Ali Cochran, Indiana University Kokomo

Ali Cochran is an undergraduate student studying mathematics at Indiana University Kokomo. She plans to pursue graduate studies to deepen her expertise in the field. Outside of academia, she runs a crochet business and enjoys recording music, blending creativity with her analytical pursuits.

Nathan Glisson, Marian University

Nathan Glisson hails from Cincinnati, Ohio. Currently a senior double majoring in physics and mechanical engineering, he will be receiving his degree in physics from Marian University and his degree in mechanical engineering from Purdue. He is a four-year member of the Marian Knights Football team and volunteers with a local youth ministry group called Str8Up Life. In his spare time, he enjoys hanging out with friends, sitting out in nature, and building with Legos.

Bryce Jennings, Indiana University Kokomo

Bryce Jennings is an undergraduate student at Indiana University Kokomo and is pursuing a double major in Mathematics and Computer Science, with a projected gradation date of May 2024. Growing up in Fort Wayne, Indiana, Bryce found his appreciation for math at Fort Wayne Snider High School when taking AP Calculus. Bryce is also a cross country and track athlete for IU Kokomo.

Katy McDicken, Marian University

Katy McDicken worked on this paper as a junior at Marian University. She is currently a 5th year in a dual degree program with Purdue University Indianapolis studying Mathematics and Mechanical Engineering. She is also on the cycling team at Marian and hopes to continue pursuing cycling and work in the engineering field after graduation.

Luke James Proctor, Marian University

Luke James Proctor graduated from Marian University with a Bachelors in Mathematics and Philosophy. He is interested in Abstract Algebra, Topology, Economic Theories, Philosophy of Science, German Philosophy, and Neo-Aristotelianism. Luke currently works at Dayton Freight as a W&R inspector and plans to work in software engineering and to eventually attend graduate school to get a PhD in Mathematics.

Sarah Klanderman, Marian University

Sarah Klanderman is an Assistant Professor of Mathematics at Marian University. Her research interests include computations related to topological Hochschild homology, studying students’ transition to proof-writing courses, connections between mathematics and other disciplines, and working with undergraduate research students. Her grant work is focused on supporting underrepresented students in STEM.

Amelia Tebbe, Indiana University Kokomo

Amelia Tebbe is an Associate Professor of Mathematics at Indiana University Kokomo. Her research interests include algebraic topology, particularly functor calculus and applications of combinatorics to homotopy theory, and working with undergraduates. Outside of work, she enjoys hiking with her dog, gardening, and knitting.

References

Richard Stanley. “Catalan Numbers” Cambridge University Press, New York (2015)

Alvarez, Bergner, Lopez. “Action Graphs and Catalan Numbers.” Journal of Integer Sequences Vol. 18 (2015) https://arxiv.org/abs/1503.00044v1

Danielle Cressman, Jonathan Lin, An Nguyen, and Luke Wiljanen. “Generalized action graphs.” (In preparation)

Julia E. Bergner and Philip Hackney. “Reedy categories which encode the notion of category actions.” Fundamenta Mathematicae 228.3 (2015) p. 193-222. http://eudml.org/doc/282637

H.W.Gould and Jocelyn Quaintance. “Combinatorial Identities: Table II: Advanced Techniques for Summing Finite Series.” https://-math.wvu.edu/ hgould/Vol.5.PDF

Online Encyclopedia of Integer Sequences, https://oeis.org/A009766

Downloads

Published

2025-03-28

How to Cite

Caldwell, D., Cochran, A., Glisson, N., Jennings, B., McDicken, K., Proctor, L. J., … Tebbe, A. (2025). Catalan Number Sequences and Generalized Action Graphs. Mathematics Exchange, 18(1), 88–106. https://doi.org/10.33043/r2y588ab

Issue

Vol. 18 No. 1 (2024): Mathematics Exchange

Section

Articles

License

Copyright (c) 2024 Drew Caldwell, Ali Cochran, Nathan Glisson, Bryce Jennings, Katy McDicken, Luke James Proctor, Sarah Klanderman, Amelia Tebbe

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Funding data