Generalizing Stern's Diatomic Array to all Restricted Partition Functions by Rene Mijares '22
Tue, May 17th, 2022
1:00 pm - 1:40 pm
Generalizing Stern’s Diatomic Array to all Restricted Partition Functions by Rene Mijares ’22, Mathematics Senior Thesis Defense, Tuesday, May 17, 1 – 1:40 pm, North Science Building 113, Wachenheim.
Abstract: Partition numbers are a well-established discipline in number theory and combinatorics, studied at length by mathematicians for centuries. A partition of a positive integer n is a way of writing n as a sum of positive integers, where two sums are considered the same partition if they differ only in the order of their summands. It is standard to denote the number of partitions of a positive integer n as p(n). There are many natural ways to restrict the partition function p(n), but we concern ourselves with the simultaneous restriction of two parameters: the integers we use as summands for partitions, and the number of times each part can be used in a partition. Specifically, we restrict the integers we use as summands for partitions to be the powers of a positive integer k > 1, and we focus the great majority of our attention on the number of repetitions d to be greater than or equal to k. We define the partition function ck(d;n), where k denotes the integer whose powers we will use, d the number of repetitions we allow for each partition (namely, d-1 repetitions), and n the integer we are partitioning.
The work done on this thesis is inspired by the extensive research on Stern’s Diatomic Sequence, the sequence generated by the function c2(3;n) when evaluated at n = 0, 1, 2, . . ., mainly that of Sam Northshield, Marjorie Bicknell-Johnson, and Bruce Reznick. This overarching goal of this thesis is to build a bridge among all restricted partition function of the type defined, and the accomplishments to that effect are two theorems, one conjecture, a generalization of the construction of Stern’s diatomic Array, and a catalog of restricted partition functions with unique recursive formulas and non-rigorous but non-trivial observations about the architecture of their corresponding array. Most of the research done on partition functions of this type has focused on growth rate and asymptotic behavior, characteristic polynomials, and linear analysis, but on this thesis, we take a new direction on the exploration of these partition functions to encompass all bases k and all number of repetitions d.