On the Structure of Bispecial Sturmian Words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of \emph{all} Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167

Factorizations of the Fibonacci Infinite Word

The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers

Italian Co-operative Law Reform and Co-operative Principles

A reform of Italian co-operative law was passed in 2003 and came into force in 2004. This paper presents the principal characteristics of the new Italian co-operative law and seeks to evaluate the relationship of some of its main provisions to traditional co-operative principles. From this perspective, the paper deals in particular with the definition of the Italian co-operative as a company with a “mutual purpose”; the distinction between “mainly mutual” co-operatives and “other” co-operatives (and the relationship between mutuality and profit-making in co-operatives); the regulation of voting in the assembly (the “one member, one vote” principle and its exceptions); the available governance systems (“tripartite”, “dualistic”, “monistic”); and co-operative finance solutions (investor members and financial instruments). Using the Italian reforms as a starting point for debate, this paper puts forth the possibility of generalising a modified approach to co-operative regulation and principles, taking into account efficiency issues, while preserving the co-operative identity. Paper presented at the symposium “Co-operatives, legislation and public policy” (Sobey School of Business, Saint Mary’s University, Halifax, Canada, 19-21 June 2009) (Nel 2003 è stata approvata in Italia e poi nel 2004 entrata in vigore una nuova riforma legislativa sulle imprese cooperative. Il presente articolo presenta le principali caratteristiche della nuova legge e tenta di valutare il rapporto tra alcuni dei principali provvedimenti stabiliti dalla legge e i principi tradizionali delle cooperative. Da questo punto di vista, l’articolo affronta in particolare la definizione di cooperativa vigente in Italia, che la inquadra come una impresa a “finalità mutualistica”. Vengono inoltre affrontati il tema della distinzione tra cooperative a mutualità prevalente e altre cooperative (e la relazione tra mutualità e raggiungimento di profitti nelle cooperative), la regolazione dei meccanismi di voto (nel rispetto del principio ‘una-testa-un-voto’ e con le sue eccezioni), i sistemi di governance (tripartito, dualistico e monistico), le soluzioni introdotte per il sostegno finanziario delle cooperative (con la figura dei soci sovventori o investitori ed alcuni strumenti finanziari). L’articolo utilizza quindi la riforma giuridica introdotta per aprire il dibattito ed allargare alla possibilità di approcciarsi alla regolamentazione e ai principi delle cooperative in un approccio nuovo che tenga conto della questione dell’efficienza pur preservando l’identità cooperativa.)Italian cooperatives, cooperative principles, mutual purpose, governance systems, corporate governance, Italian law

Vertical representation of C∞C^{\infty}-words

We present a new framework for dealing with $C^{\infty}$-words, based on their left and right frontiers. This allows us to give a compact representation of them, and to describe the set of $C^{\infty}$-words through an infinite directed acyclic graph $G$. This graph is defined by a map acting on the frontiers of $C^{\infty}$-words. We show that this map can be defined recursively and with no explicit references to $C^{\infty}$-words. We then show that some important conjectures on $C^{\infty}$-words follow from analogous statements on the structure of the graph $G$.Comment: Published in Theoretical Computer Scienc

On the least number of palindromes contained in an infinite word

We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.Comment: Accepted for publication in Theoretical Computer Scienc

Open and Closed Prefixes of Sturmian Words

A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of reversed standard words.Comment: To appear in WORDS 2013 proceeding

Abelian-Square-Rich Words

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ 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 $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-square-rich} if every factor of $w$ 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 $2n$ of the Thue-Morse word is $2$-regular. As for Sturmian words, we prove that a Sturmian word $s_{\alpha}$ of angle $\alpha$ is uniformly abelian-square-rich if and only if the irrational $\alpha$ has bounded partial quotients, that is, if and only if $s_{\alpha}$ has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the proof of Proposition

On the Minimal Uncompletable Word Problem

Let S be a finite set of words over an alphabet Sigma. The set S is said to be complete if every word w over the alphabet Sigma is a factor of some element of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are interested in finding bounds on the minimal length of words in Sigma* which are not elements of Fact(S*) in terms of the maximal length of words in S.Comment: 5 pages; added references, corrected typo

The sequence of open and closed prefixes of a Sturmian word

A finite word is closed if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We are interested in the {\it oc-sequence} of a word, which is the binary sequence whose $n$-th element is $0$ if the prefix of length $n$ of the word is open, or $1$ if it is closed. We exhibit results showing that this sequence is deeply related to the combinatorial and periodic structure of a word. In the case of Sturmian words, we show that these are uniquely determined (up to renaming letters) by their oc-sequence. Moreover, we prove that the class of finite Sturmian words is a maximal element with this property in the class of binary factorial languages. We then discuss several aspects of Sturmian words that can be expressed through this sequence. Finally, we provide a linear-time algorithm that computes the oc-sequence of a finite word, and a linear-time algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of arXiv:1306.225

Enumeration and Structure of Trapezoidal Words

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, \emph{semicentral words}, and show that they are characterized by the property that they can be written as $uxyu$, for a central word $u$ and two different letters $x,y$. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.Comment: Accepted for publication in Theoretical Computer Scienc