195 research outputs found
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
Generalized trapezoidal words
The factor complexity function of a finite or infinite word
counts the number of distinct factors of of length for each .
A finite word of length is said to be trapezoidal if the graph of its
factor complexity as a function of (for ) is
that of a regular trapezoid (or possibly an isosceles triangle); that is,
increases by 1 with each on some interval of length , then
is constant on some interval of length , and finally
decreases by 1 with each on an interval of the same length . Necessarily
(since there is one factor of length , namely the empty word), so
any trapezoidal word is on a binary alphabet. Trapezoidal words were first
introduced by de Luca (1999) when studying the behaviour of the factor
complexity of finite Sturmian words, i.e., factors of infinite "cutting
sequences", obtained by coding the sequence of cuts in an integer lattice over
the positive quadrant of made by a line of irrational slope.
Every finite Sturmian word is trapezoidal, but not conversely. However, both
families of words (trapezoidal and Sturmian) are special classes of so-called
"rich words" (also known as "full words") - a wider family of finite and
infinite words characterized by containing the maximal number of palindromes -
studied in depth by the first author and others in 2009.
In this paper, we introduce a natural generalization of trapezoidal words
over an arbitrary finite alphabet , called generalized trapezoidal
words (or GT-words for short). In particular, we study combinatorial and
structural properties of this new class of words, and we show that, unlike the
binary case, not all GT-words are rich in palindromes when , but we can describe all those that are rich.Comment: Major revisio
Palindromic richness for languages invariant under more symmetries
For a given finite group consisting of morphisms and antimorphisms of a
free monoid , we study infinite words with language closed under
the group . We focus on the notion of -richness which describes words
rich in generalized palindromic factors, i.e., in factors satisfying
for some antimorphism . We give several
equivalent descriptions which are generalizations of know characterizations of
rich words (in the terms of classical palindromes) and show two examples of
-rich words
On morphisms preserving palindromic richness
It is known that each word of length contains at most distinct
palindromes. A finite rich word is a word with maximal number of palindromic
factors. The definition of palindromic richness can be naturally extended to
infinite words. Sturmian words and Rote complementary symmetric sequences form
two classes of binary rich words, while episturmian words and words coding
symmetric -interval exchange transformations give us other examples on
larger alphabets. In this paper we look for morphisms of the free monoid, which
allow to construct new rich words from already known rich words. We focus on
morphisms in Class . This class contains morphisms injective on the
alphabet and satisfying a particular palindromicity property: for every
morphism in the class there exists a palindrome such that
is a first complete return word to for each letter . We
characterize morphisms which preserve richness over a binary
alphabet. We also study marked morphisms acting on alphabets with
more letters. In particular we show that every Arnoux-Rauzy morphism is
conjugated to a morphism in Class and that it preserves richness
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
Rich Words and Balanced Words
This thesis is mostly focused on palindromes. Palindromes have been studied extensively, in recent years, in the field of combinatorics on words.Our main focus is on rich words, also known as full words. These are words which have maximum number of distinct palindromes as factors.We shed some more light on these words and investigate certain restricted problems.
Finite rich words are known to be extendable to infinite rich words. We study more closely how many different ways, and in which situations, rich words can be extended so that they remain rich.The defect of a ord is defined to be the number of palindromes the word is lacking.We will generalize the definition of defect with respect to extending the word to be infinite.The number of rich words, on an alphabet of size , is given an upper and a lower bound.
Hof, Knill and Simon presented (Commun. Math. Phys. 174, 1995) a well-known question whether all palindromic subshifts which are enerated by primitive substitutions arise from substitutions which are in class P. Over the years, this question has transformed a bit and is nowadays called the class P conjecture. The main point of the conjecture is to attempt to explain how an infinite word can contain infinitely many palindromes.We will prove a partial result of the conjecture.
Rich square-free words are known to be finite (Pelantov\'a and Sarosta, Discrete Math. 313, 2013). We will give another proof for that result. Since they are finite, there exists a longest such word on an -ary alphabet.We give an upper and a lower bound for the length of that word.
We study also balanced words. Oliver Jenkinson proved (Discrete Math., Alg. and Appl. 1(4), 2009) that if we take the partial sum of the lexicographically ordered orbit of a binary word, then the balanced word gives the least partial sum. The balanced word also gives the largest product. We will show that, at the other extreme, there are the words of the form ( and are integers with ), which we call the most unbalanced words. They give the greatest partial sum and the smallest product.Tässä väitÜskirjassa käsitellään pääasiassa palindromeja. Palindromeja on tutkittu viime vuosina runsaasti sanojen kombinatoriikassa.Suurin kiinnostuksen kohde tässä tutkielmassa on rikkaissa sanoissa. Nämä ovat sanoja
joissa on maksimaalinen määrä erilaisia palindromeja tekijÜinä.Näitä sanoja tutkitaan monesta eri näkÜkulmasta.
Ăärellisiä rikkaita sanoja voidaan tunnetusti jatkaa äärettĂśmiksi rikkaiksi sanoiksi.TyĂśssä tutkitaan tarkemmin sitä, miten monella tavalla ja missä eri tilanteissa rikkaita sanoja voidaan jatkaa siten, että ne pysyvät rikkaina.Sanan vajauksella tarkoitetaan puuttuvien palindromien lukumäärää.Vajauksen käsite yleistetään tapaukseen, jossa sanaa on jatkettava äärettĂśmäksi sanaksi.Rikkaiden sanojen lukumäärälle annetaan myĂśs ylä- ja alaraja.
Hof, Knill ja Simon esittivät kysymyksen (Commun. Math. Phys. 174, 1995), saadaanko kaikki äärettÜmät sanat joissa on ääretÜn määrä palindromeja tekijÜinä ja jotka ovat primitiivisen morfismin generoimia, morfismeista jotka kuuluvat luokkaan P. Nykyään tätä ongelmaa kutsutaan luokan P konjektuuriksi ja sen tarkoitus on saada selitys sille,millä tavalla äärettÜmässä sanassa voi olla tekijÜinä äärettÜmän monta palindromia. Osittainen tulos tästä konjektuurista todistetaan.
Rikkaiden neliÜvapaiden sanojen tiedetään olevan äärellisiä (Pelantov\'a ja Starosta, Discrete Math. 313, 2013).
Tälle tulokselle annetaan uudenlainen todistus.Koska kyseiset sanat ovat äärellisiä, voidaan selvittää mikä niistä on pisin.Ylä- ja alaraja annetaan tällaisen pisimmän sanan pituudelle.
TyÜssä tutkitaan myÜs tasapainotettuja sanoja.Tasapainotetut sanat antavat pienimmän osittaissumman binäärisille sanoille (Jenkinson, Discrete Math., Alg. and Appl. 1(4), 2009).Lisäksi ne antavat suurimman tulon.Muotoa ( ja ovat kokonaislukuja joille ) olevien sanojen todistetaan vastaavasti antavan suurimman osittaissumman ja pienimmän tulon.Ne muodostavat täten toisen ääripään tasapainotetuille sanoille, ja asettavat kaikki muut sanat näiden väliin.Siirretty Doriast
- âŚ