Asymptotic Enumeration of k-bounded Functions Using Toeplitz Matrices

Authors

• Tiago Cavalcante Trindade, Alpha Lumen InstituteFollow
• Pedro Martineli, Alpha Lumen InstituteFollow

Abstract

We introduce the concept of $(\alpha, \lambda)$-bounded functions, characterized by limited local variation, which is especially useful in discrete sets. Initially, we formally define these functions and investigate their fundamental properties, highlighting significant differences from continuous functions. The main result obtained is the asymptotic estimate of $a(n, k)$, representing the number of functions from $[n]$ to $[n]$ that are $k$-bounded with respect to the Manhattan distance. The proof of this result combines Toeplitz matrices with a well-known inequality from graph theory.

Author Bio

Tiago Trindade is a 12th-grade student at Alpha Lumen Institute. He is passionate about math olympiads and problem-solving. Outside of math, he enjoys discovering cult-classic films. He plans to major in mathematics.

Pedro Martineli is a 12th-grade student at Alpha Lumen Institute. Having started his coding journey at the age of 10, he is passionate about low-level programming, exploring complex topics. Beyond the world of code and math, he is an avid aviation enthusiast who enjoys piloting aircraft. Plans to major in computer science.

Faculty Sponsor

Lucas Henrique Martins da Silva

Recommended Citation

Trindade, Tiago Cavalcante and Martineli, Pedro (2025) "Asymptotic Enumeration of k-bounded Functions Using Toeplitz Matrices," Rose-Hulman Undergraduate Mathematics Journal: Vol. 26: Iss. 2, Article 7.
Available at: https://scholar.rose-hulman.edu/rhumj/vol26/iss2/7