|dc.date.accessioned||2022-10-12 13:19:19 (GMT)||
|dc.date.available||2022-10-12 13:19:19 (GMT)||
|dc.description.abstract||Combinatorics on words is a field of mathematics and theoretical computer science that
is concerned with sequences of symbols called words, or strings. One class of words that
are ubiquitous in combinatorics on words, and theoretical computer science more broadly,
are the bordered words. The word w has a border u if u is a non-empty proper prefix and
suffix of w. The word w is said to be bordered if it has a border. Otherwise w is said to
This thesis is primarily concerned with variations and generalizations of bordered and
In Chapter 1 we introduce the field of combinatorics on words and give a brief overview
of the literature on borders relevant to this thesis.
In Chapter 2 we give necessary definitions, and we present a more in-depth literature
review on results on borders relevant to this thesis.
In Chapter 3 we complete the characterization due to Harju and Nowotka of binary
words with the maximum number of unbordered conjugates. We also show that for every
number, up to this maximum, there exists a binary word with that number of unbordered
In Chapter 4 we give results on pairs of words that almost commute and anti-commute.
Two words x and y almost commute if xy and yx differ in exactly two places, and they
anti-commute if xy and yx differ in all places. We characterize and count the number of
pairs of words that almost and anti-commute. We also characterize and count variations
of almost-commuting words. Finally we conclude with some asymptotic results related to
the number of almost-commuting pairs of words.
In Chapter 5 we count the number of length-n bordered words with a unique border.
We also show that the probability that a length-n word has a unique border tends to a
In Chapter 6 we present results on factorizations of words related to borders, called
block palindromes. A block palindrome is a factorization of a word into blocks that turns
into a palindrome if each identical block is replaced by a distinct character. Each block is a
border of a central block. We call the number of blocks in a block palindrome the width of
the block palindrome. The largest block palindrome of a word is the block palindrome of the
word with the maximum width. We count all length-n words that have a width-t largest
block palindrome. We also show that the expected width of a largest block palindrome
tends to a constant. Finally we conclude with some results on another extremal variation
of block palindromes, the smallest block palindrome.
In Chapter 7 we present the main results of the thesis. Roughly speaking, a word is
said to be closed if it contains a non-empty proper border that occurs exactly twice in the
word. A word is said to be privileged if it is of length ≤ 1 or if it contains a non-empty
proper privileged border that occurs exactly twice in the word. We give new and improved
bounds on the number of length-n closed and privileged words over a k-letter alphabet.
In Chapter 8 we work with a generalization of bordered words to pairs of words. The
main result of this chapter is a characterization and enumeration result for this generalization
of bordered words to multiple dimensions.
In Chapter 9 we conclude by summarizing the results of this thesis and presenting
avenues for future research.||en
|dc.publisher||University of Waterloo||en
|dc.title||On the Properties and Structure of Bordered Words and Generalizations||en
|uws-etd.degree.department||David R. Cheriton School of Computer Science||en
|uws-etd.degree.grantor||University of Waterloo||en
|uws-etd.degree||Doctor of Philosophy||en
|uws.contributor.affiliation1||Faculty of Mathematics||en