#### Date of Award

5-2012

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Legacy Department

Mathematics

#### Advisor

Calkin, Neil

#### Committee Member

Gallagher , Colin

#### Committee Member

James , Kevin

#### Committee Member

Matthews , Gretchen

#### Abstract

We present several problems in combinatorics including the partition function, Graph Nim, and the evolution of strings.

Let p(n) be the number of partitions of n. We say a sequence a_{n} is log-concave if for every n, a_{n}2 &ge a_{n+1} a_{n-1}. We will show that p(n) is log-concave for n &ge 26. We will also show that for n<26, p(n) alternatively satisfies and does not satisfy the log-concave property. We include results for the Sperner property of the partition function.

The second problem we present is the game of Graph Nim. We use the Sprague-Grundy theorem to analyze modified versions of Nim played on various graphs. We include progress made towards proving that all G-paths are periodic.

The third topic we present is on the evolution of strings. Consider a string of length l over an alphabet of size k. At each stage of the evolution, with a probability of q, we randomly select a new letter to replace a correct letter. Using the transition matrix we will study the absorption rate to the correct string.

#### Recommended Citation

Janoski, Janine, "A Collection of Problems in Combinatorics" (2012). *All Dissertations*. 892.

https://tigerprints.clemson.edu/all_dissertations/892