Group Theoretical Approach to partitioning of Integers

This is a short, upper-level undergraduate/early graduate mini-course exploring the deep connection between integer partitions, symmetry, and group theory. The course is based on the paper L.K. Mork, K. Sullivan, T. Vogt, and D.J. Ulness "A group theoretical approach to partitioning of integers: Application to triangular numbers, squares, and centered polygonal numbers" Australasian Journal of Combinatorics 80(3) (2021), 305-321. Building from combinatorial intuition toward the representation theory of the symmetric group, the course develops a structured pathway for understanding how generating functions, Young diagrams, and irreducible representations illuminate the structure of m-colored partitions and figurate-number partitions. Students will learn both the classical combinatorics and the group-theoretical viewpoint introduced in our published work, while gaining hands-on experience through the accompanying Python application designed for exploratory mathematics. This course is ideal for students in mathematics, theoretical chemistry, and mathematical physics who want to see how algebraic symmetry organizes and explains partition structures.

Resources

Main paper: 

L.K. Mork, K. Sullivan, T. Vogt, and D.J. Ulness "A group theoretical approach to partitioning of integers: Application to triangular numbers, squares, and centered polygonal numbers," Australasian Journal of Combinatorics 80(3) (2021), 305-321.

Secondary paper:

M. D. Hirschhorn and J. A. Sellers, "Partitions into three triangular numbers," Australasian Journal of Combinatorics 30 (2004), 307–318.

Software

The zip files below contain the installer and an installation guide for FiguratePartitionExplorer.

• Windows

• Mac

• Linux (AppImage)

The GitHub Repository is here.