273 research outputs found
A Characterization of Infinite LSP Words
G. Fici proved that a finite word has a minimal suffix automaton if and only
if all its left special factors occur as prefixes. He called LSP all finite and
infinite words having this latter property. We characterize here infinite LSP
words in terms of -adicity. More precisely we provide a finite set of
morphisms and an automaton such that an infinite word is LSP if
and only if it is -adic and all its directive words are recognizable by
A Coloring Problem for Sturmian and Episturmian Words
We consider the following open question in the spirit of Ramsey theory: Given
an aperiodic infinite word , does there exist a finite coloring of its
factors such that no factorization of is monochromatic? We show that such a
coloring always exists whenever is a Sturmian word or a standard
episturmian word
k-Spectra of weakly-c-Balanced Words
A word is a scattered factor of if can be obtained from by
deleting some of its letters. That is, there exist the (potentially empty)
words , and such that and
. We consider the set of length- scattered
factors of a given word w, called here -spectrum and denoted
\ScatFact_k(w). We prove a series of properties of the sets \ScatFact_k(w)
for binary strictly balanced and, respectively, -balanced words , i.e.,
words over a two-letter alphabet where the number of occurrences of each letter
is the same, or, respectively, one letter has -more occurrences than the
other. In particular, we consider the question which cardinalities n=
|\ScatFact_k(w)| are obtainable, for a positive integer , when is
either a strictly balanced binary word of length , or a -balanced binary
word of length . We also consider the problem of reconstructing words
from their -spectra
Avoidability of formulas with two variables
In combinatorics on words, a word over an alphabet is said to
avoid a pattern over an alphabet of variables if there is no
factor of such that where is a
non-erasing morphism. A pattern is said to be -avoidable if there exists
an infinite word over a -letter alphabet that avoids . We consider the
patterns such that at most two variables appear at least twice, or
equivalently, the formulas with at most two variables. For each such formula,
we determine whether it is -avoidable, and if it is -avoidable, we
determine whether it is avoided by exponentially many binary words
Pairwise Well-Formed Modes and Transformations
One of the most significant attitudinal shifts in the history of music
occurred in the Renaissance, when an emerging triadic consciousness moved
musicians towards a new scalar formation that placed major thirds on a par with
perfect fifths. In this paper we revisit the confrontation between the two
idealized scalar and modal conceptions, that of the ancient and medieval world
and that of the early modern world, associated especially with Zarlino. We do
this at an abstract level, in the language of algebraic combinatorics on words.
In scale theory the juxtaposition is between well-formed and pairwise
well-formed scales and modes, expressed in terms of Christoffel words or
standard words and their conjugates, and the special Sturmian morphisms that
generate them. Pairwise well-formed scales are encoded by words over a
three-letter alphabet, and in our generalization we introduce special positive
automorphisms of , the free group over three letters.Comment: 12 pages, 3 figures, paper presented at the MCM2017 at UNAM in Mexico
City on June 27, 2017, keywords: pairwise well-formed scales and modes,
well-formed scales and modes, well-formed words, Christoffel words, standard
words, central words, algebraic combinatorics on words, special Sturmian
morphism
Nivat's conjecture holds for sums of two periodic configurations
Nivat's conjecture is a long-standing open combinatorial problem. It concerns
two-dimensional configurations, that is, maps where is a finite set of symbols. Such configurations are often
understood as colorings of a two-dimensional square grid. Let denote
the number of distinct block patterns occurring in a configuration
. Configurations satisfying for some
are said to have low rectangular complexity. Nivat conjectured that such
configurations are necessarily periodic.
Recently, Kari and the author showed that low complexity configurations can
be decomposed into a sum of periodic configurations. In this paper we show that
if there are at most two components, Nivat's conjecture holds. As a corollary
we obtain an alternative proof of a result of Cyr and Kra: If there exist such that , then is periodic. The
technique used in this paper combines the algebraic approach of Kari and the
author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with
proofs. 12 pages + references + appendi
On the maximal sum of exponents of runs in a string
A run is an inclusion maximal occurrence in a string (as a subinterval) of a
repetition with a period such that . The exponent of a run
is defined as and is . We show new bounds on the maximal sum of
exponents of runs in a string of length . Our upper bound of is
better than the best previously known proven bound of by Crochemore &
Ilie (2008). The lower bound of , obtained using a family of binary
words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the
maximal sum of exponents of runs in a string of length is smaller than Comment: 7 pages, 1 figur
Enumerating Abelian Returns to Prefixes of Sturmian Words
We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian
returns in Sturmian words. We determine the cardinality of the set
of abelian returns of all prefixes of a Sturmian word in
terms of the coefficients of the continued fraction of the slope, dependingly
on the intercept. We provide a simple algorithm for finding the set
and we determine it for the characteristic Sturmian words.Comment: 19page
Palindromic complexity of trees
We consider finite trees with edges labeled by letters on a finite alphabet
. Each pair of nodes defines a unique labeled path whose trace is a
word of the free monoid . The set of all such words defines the
language of the tree. In this paper, we investigate the palindromic complexity
of trees and provide hints for an upper bound on the number of distinct
palindromes in the language of a tree.Comment: Submitted to the conference DLT201
Detecting One-variable Patterns
Given a pattern such that
, where is a
variable and its reversal, and
are strings that contain no variables, we describe an
algorithm that constructs in time a compact representation of all
instances of in an input string of length over a polynomially bounded
integer alphabet, so that one can report those instances in time.Comment: 16 pages (+13 pages of Appendix), 4 figures, accepted to SPIRE 201
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