A Collection of Problems in Combinatorics

Author

Janine Janoski, Clemson UniversityFollow

Date of Award

5-2012

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematics

Committee Chair/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_n2 &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