Mathematics Senior Thesis Defense by Matthew Phang '23

Obtaining Integer Partitions Using the Selmer and Brun Algorithms by Matthew Phang ’23, Mathematics Senior Thesis Defense, Friday, May 5, 1 – 1:45 pm, North Science Building 015, Wachenheim.

Abstract:  Integer partitions and multidimensional continued fraction algorithms have been studied at length within their respective fields.  In this thesis, we are interested in the intersection of these two areas.  Multidimensional continued fraction algorithms such as the triangle map have previously been shown to act on integer partitions as a way of mapping given integer partitions to other partitions of that same integer.  We focus on how the Selmer and Brun algorithms act analogously on integer partitions.  We first study general properties of the Selmer and Brun maps.  Previous work has been done on repeatedly applying other maps to initial integer partitions to obtain sequences of distinct integer partitions.  We show that the Selmer and Brun maps can be used repeatedly to obtain any possible partition of n starting with an initial partition of only a single part. Interestingly, this property is not shared by all multidimensional continued fraction algorithms.