Date of Award
Doctor of Philosophy (PhD)
Gallagher , Colin
James , Kevin
Matthews , Gretchen
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 an is log-concave if for every n, an2 &ge an+1 an-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.
Janoski, Janine, "A Collection of Problems in Combinatorics" (2012). All Dissertations. 892.