Date of Degree

9-2015

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Melvyn Nathanson

Subject Categories

Mathematics

Keywords

Eulerian; Hyperbinary; Integer Partitions; Number Theory; Partitions; Stern

Abstract

The first chapter examines $p_b(n)$, the number of partitions of $n$ into powers of $b$, along with a family of identities which can be deduced by iterating a recurrence satisfied by $p_b(n)$ in a suitable way. These identities can then be used to calculate $p_b(n)$ for large values of $n$.

The second chapter restricts these types of partitions even further, limiting the multiplicity of each part. Its object of study is $p_{b,d}(n)$, that is, the number of partitions of $n$ into powers of $b$ repeating each power at most $d$ times. The methods of the first chapter are applied, and the self-similarity of these sequences is discussed in detail.

The third chapter focuses on $p_{A,M}(n)$, the number of partitions of $n$ with parts in $A$ and multiplicities in $M$. A construction of Alon which produces infinite sets $A$ and $M$ so that $p_{A,M}(n) = 1$ is generalized so that $A$ can be chosen to be a subset of powers of a given base.

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