Counting convex shapes
Date
2024
Authors
Wise, Elitza
Journal Title
Journal ISSN
Volume Title
Publisher
University of Victoria
Abstract
This research project aimed to examine C(n), the number of convex connected subsets of lattice containing n points. The primary objective was to derive and analyse an upper bound for C(n) to determine whether it grows sub-exponentially. This was done by programming a recursive code that took n as input and constructed every convex connected shape row-by-row. To ensure the results of C(n) were correct, the shapes were visually printed as output and further studied. This included focusing on pairs of adjacent C(n) and C(n+1) values, and confirming that the number of ways to add a point to the n-shapes was equal to the number of ways to take away a point from the (n+1)-shapes. Using a log-log regression transformation it was confirmed that C(n) is, in fact, sub-exponential. This plays a role in problems in statistical mechanics.
Description
Keywords
convexity, enumeration, asymptotics, lattice, recursion, sub-exponential growth