#### Date of Award

Fall 12-2016

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematics

#### First Advisor

Timothy B. Flowers

#### Second Advisor

Rachelle R. Bouchat

#### Third Advisor

Kimberly J. Burch

#### Abstract

In this thesis we will review the beginnings of integer partitions. We will further explore past research done on restricted partition functions, specifically those of the m-ary and hyper-m-ary partitions for m ≥ 2. The main content of this paper explores the kth m–ary partition function. For m, k ≥ 2, let the kth m–ary partition function, bm(k, n), be the number of ways we can write a positive integer n as a sum of powers of m using at most k of each power. In this thesis we will classify the monotonicity properties of bm(k, n) by considering the congruence classes of n and k modulo m, and define a relationship between bm(1, n) and writing n in base m.

#### Recommended Citation

Rucci, Laura E., "The Kth M-ary Partition Function" (2016). *Theses and Dissertations (All)*. 1431.

http://knowledge.library.iup.edu/etd/1431