500 research outputs found
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Binary words avoiding xx^Rx and strongly unimodal sequences
In previous work, Currie and Rampersad showed that the growth of the number
of binary words avoiding the pattern xxx^R was intermediate between polynomial
and exponential. We now show that the same holds for the growth of the number
of binary words avoiding the pattern xx^Rx. Curiously, the analysis for xx^Rx
is much simpler than that for xxx^R. We derive our results by giving a
bijection between the set of binary words avoiding xx^Rx and a class of
sequences closely related to the class of "strongly unimodal sequences."Comment: 4 page
Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences
We prove that the property of being closed (resp., palindromic, rich,
privileged trapezoidal, balanced) is expressible in first-order logic for
automatic (and some related) sequences. It therefore follows that the
characteristic function of those n for which an automatic sequence x has a
closed (resp., palindromic, privileged, rich, trape- zoidal, balanced) factor
of length n is automatic. For privileged words this requires a new
characterization of the privileged property. We compute the corresponding
characteristic functions for various famous sequences, such as the Thue-Morse
sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the
period-doubling sequence, and the Fibonacci sequence. Finally, we also show
that the function counting the total number of palindromic factors in a prefix
of length n of a k-automatic sequence is not k-synchronized
Growth rate of binary words avoiding
Consider the set of those binary words with no non-empty factors of the form
. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words
grows polynomially or exponentially with length. In this paper, we demonstrate
the existence of upper and lower bounds on the number of such words of length
, where each of these bounds is asymptotically equivalent to a (different)
function of the form , where , are constants
Decision Algorithms for Ostrowski-Automatic Sequences
We extend the notion of automatic sequences to a broader class, the Ostrowski-automatic sequences. We develop a procedure for computationally deciding certain combinatorial and enumeration questions about such sequences that can be expressed as predicates in first-order logic.
In Chapter 1, we begin with topics and ideas that are preliminary to this work, including a small introduction to non-standard positional numeration systems and the relationship between words and automata. In Chapter 2, we define the theoretical foundations for recognizing addition in a generalized Ostrowski numeration system and formalize the general theory that develops our decision procedure. Next, in Chapter 3, we show how to implement these ideas in practice, and provide the implementation as an integration to the automatic theorem-proving software package -- Walnut.
Further, we provide some applications of our work in Chapter 4. These applications span several topics in combinatorics on words, including repetitions, pattern-avoidance, critical exponents of special classes of words, properties of Lucas words, and so forth. Finally, we close with open problems on decidability and higher-order numeration systems and discuss future directions for research
- …