9 research outputs found
A construction for variable dimension strong non-overlapping matrices
We propose a method for the construction of sets of variable dimension strong
non-overlapping matrices basing on any strong non-overlapping set of strings
Restricting Dyck Paths and 312-avoiding Permutations
Dyck paths having height at most and without valleys at height are
combinatorially interpreted by means of 312-avoding permutations with some
restrictions on their \emph{left-to-right maxima}. The results are obtained by
analyzing a restriction of a well-known bijection between the sets of Dyck
paths and 312-avoding permutations. We also provide a recursive formula
enumerating these two structures using ECO method and the theory of production
matrices. As a further result we obtain a family of combinatorial identities
involving Catalan numbers
Convergence of the Number of Period Sets in Strings
Consider words of length n. The set of all periods of a word of length n is a subset of {0, 1, 2, . . ., n−1}. However, any subset of {0, 1, 2, . . ., n−1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko proposed to encode the set of periods of a word into an n long binary string, called an autocorrelation, where a one at position i denotes the period i. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for strings of length n, denoted κn. They conjectured that ln(κn) asymptotically converges to a constant times ln2(n). Although improved lower bounds for ln(κn)/ln2(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzko’s paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings
Constructions and bounds for codes with restricted overlaps
Non-overlapping codes have been studied for almost 60 years. In such a code,
no proper, non-empty prefix of any codeword is a suffix of any codeword. In
this paper, we study codes in which overlaps of certain specified sizes are
forbidden. We prove some general bounds and we give several constructions in
the case of binary codes. Our techniques also allow us to provide an
alternative, elementary proof of a lower bound on non-overlapping codes due to
Levenshtein in 1964.Comment: 25 pages. Extra citations, typos corrected and explanations expande
On non-expandable cross-bifix-free codes
A cross-bifix-free code of length over is defined as a
non-empty subset of satisfying that the prefix set of each
codeword is disjoint from the suffix set of every codeword. Cross-bifix-free
codes have found important applications in digital communication systems. One
of the main research problems on cross-bifix-free codes is to construct
cross-bifix-free codes as large as possible in size. Recently, Wang and Wang
introduced a family of cross-bifix-free codes , which is a
generalization of the classical cross-bifix-free codes studied early by
Lvenshtein, Gilbert and Chee {\it et al.}. It is known that
is nearly optimal in size and is non-expandable if
or . In this paper, we first show that is
non-expandable if and only if or , thereby improving the
results in [Chee {\it et al.}, IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT,
2022]. We then construct a new family of cross-bifix-free codes
to expand such that the resulting larger
code is a non-expandable
cross-bifix-free code whenever is expandable. Finally, we
present an explicit formula for the size of .Comment: This paper has been submitted to IEEE T-IT for possible publicatio
On the Properties and Structure of Bordered Words and Generalizations
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
be unbordered.
This thesis is primarily concerned with variations and generalizations of bordered and
unbordered words.
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
conjugates.
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
constant.
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
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum