A note on the number of squares in a word

Fraenkel and Simpson [A.S. Fraenkel, J. Simpson, How many squares can a string contain? J. Combin. Theory Ser. A 82 (1998) 112–120] proved that the number of squares in a word of length n$n$ is bounded by 2n$2n$. In this note we improve this bound to 2n−Θ(logn)$2n-\Theta (logn)$. Based on the numerical evidence, the conjectured bound is n$n$.     

Avoiding large squares in infinite binary words

We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with $\left|y\right|⩾4$; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except $0^2$, $1^2$, and $(01{)}^2$; our construction is somewhat simpler than the original construction of Fraenkel and Simpson. In both cases, we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties. Finally, we answer an open question of Prodinger and Urbanek from 1979 by demonstrating the existence of two infinite binary words, each avoiding arbitrarily large squares, such that their perfect shuffle has arbitrarily large squares.     

Balancing and clustering of words in the Burrows-Wheeler transform

Compression algorithms based on Burrows-Wheeler transform (BWT) take advantage of the fact that the word output of BWT shows a local similarity and then turns out to be highly compressible. The aim of the present paper is to study such “clustering effect” by using notions and methods from Combinatorics on Words.The notion of balance of a word plays a central role in our investigation. Empirical observations suggest that balance is actually the combinatorial property of input word that ensure optimal BWT compression. Moreover, it is reasonable to assume that the more balanced the input word is, the more local similarity we have after BWT (and therefore the better the compression is). This hypothesis is here corroborated by experiments on “real” text, by using local entropy as a measure of the degree of balance of a word.In the setting of Combinatorics on Words, a sound confirmation of previous hypothesis is given by a result of Mantaci et al. (2003) [27], which states that, in the case of a binary alphabet, there is an equivalence between circularly balanced words, words having a clusterized BWT, and the conjugates of standard words. In the case of alphabets of size greater than two, there is no more equivalence. The last section of the present paper is devoted to investigate the relationships between these notions, and other related ones (as, for instance, palindromic richness) in the case of a general alphabet.     

Unavoidable Sets of Partial Words

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of “do not know” symbols or “holes,” appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a k-letter alphabet. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0452020. Part of this paper was presented at DLT’07 [4]. We thank the referees as well as Robert Mercaş and Geoffrey Scott for very valuable comments and suggestions. World Wide Web server interfaces have been established at and for automated use of the programs.     

Fewest repetitions versus maximal-exponent powers in infinite binary words

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give some new results on the trade-off between the number of squares and the number of maximal-exponent powers in infinite binary words, in the three cases where the maximal exponent is 7/3, 5/2, and 3. These are the only threshold values related to the question.     

On the number of Dejean words over alphabets of 5, 6, 7, 8, 9 and 10 letters

We give lower bounds on the growth rate of Dejean words, i.e. minimally repetitive words, over a k-letter alphabet, for 5≤k≤10. Put together with the known upper bounds, we estimate these growth rates with the precision of 0.005. As a consequence, we establish the exponential growth of the number of Dejean words over a k-letter alphabet, for 5≤k≤10.     

Some characterizations of Parikh matrix equivalent binary words

We state different characterizations of pair of words having the same Parikh matrix.     

Computation of words satisfying the “rhythmic oddity property” (after Simha Arom's works)

This paper addresses the problem of enumerating all words having a combinatoric property called “rhythmic oddity property”. This enumeration is motivated by the fact that this property is satisfied by many rhythmic patterns used in traditional Central African music.     

Optimal parallel detection of squares in strings

A stringw isprimitive if it is not a power of another string (i.e., writingw =v ^k impliesk = 1. Conversely,w is asquare ifw =vv, withv a primitive string. A stringx issquare-free if it has no nonempty substring of the formww. It is shown that the square-freedom of a string ofn symbols over an arbitrary alphabet can be tested by a CRCW PRAM withn processors inO(logn) time and linear auxiliary space. If the cardinality of the input alphabet is bounded by a constant independent of the input size, then the number of processors can be reduced ton/logn without affecting the time complexity of this strategy. The fastest sequential algorithms solve this problemO(n logn) orO(n) time, depending on whether the cardinality of the input alphabet is unbounded or bounded, and either performance is known to be optimal within its class. More elaborate constructions lead to a CRCW PRAM algorithm for detecting, within the samen-processors bounds, all positioned squares inx in timeO(logn) and using linear auxiliary space. The fastest sequential algorithms solve this problem inO(n logn) time, and such a performance is known to be optimal.This research was supported, through the Leonardo Fibonacci Institute, by the Istituto Trentino di Cultura, Trento, Italy. Additional support was provided by the French and Italian Ministries of Education, by the National Research Council of Italy, by the British Research Council Grant SERC-E76797, by NSF Grant CCR-89-00305, by NIH Library of Medicine Grant ROI LM05118, by AFOSR Grant 90-0107, and by NATO Grant CRG900293.     

Reconstruction of a word from a multiset of its factors

Let be the multiset containing all factors of w of length k including repetitions. One of the main results is that if for all , then w=v. The bound is optimal; however we will also show that if for all , then w and v are structurally similar.     

Square-free partial words

We say that a partial word w over an alphabet A is square-free if every factor xx^′ of w such that x and x^′ are compatible is either of the form ?a or a? where ? is a hole and a∈A. We prove that there exist uncountably many square-free partial words over a ternary alphabet with an infinite number of holes.     

A cyclic binary morphism avoiding Abelian fourth powers

We exhibit a cyclic binary morphism avoiding Abelian fourth powers.     

Covering numbers of rotations

For an irrational rotation, we use the symbolic dynamics on the sturmian coding to compute explicitly, according to the continued fraction approximation of the argument, the measure of the largest Rokhlin stack made with intervals, and the measure of the largest Rokhlin stack whose levels have one name for the coding. Each one of these measures is equal to one if and only if the argument has unbounded partial quotients.     

On two-sided infinite fixed points of morphisms

Let Σ be a finite alphabet, and let h:Σ^* → Σ^* be a morphism. Finite and infinite fixed points of morphisms—i.e., those words w such that h(w)=w—play an important role in formal language theory. Head characterized the finite fixed points of h, and later, Head and Lando characterized the one-sided infinite fixed points of h. Our paper has two main results. First, we complete the characterization of fixed points of morphisms by describing all two-sided infinite fixed points of h, for both the “pointed” and “unpointed” cases. Second, we completely characterize the solutions to the equation h(xy)=yx in finite words.