Counting Overlap-Free Binary Words

A word on a nite alphabet A is said to be overlap free if it contains no factor of the form xuxux where x is a letter and u a possibly empty word In this paper we study the number un of overlap free binary words of length n which is known to be bounded by a polynomial in n First we describe a bijection between the set of overlap free words and a rational language This yields recurrence relations for un which allow to compute un in logarithmic time Then we prove that the numbers sup f r j n O un g and inf f r j un O n g are distinct and we give an upper bound for and a lower bound for Finally we compute an asymptotically tight bound to the number of overlap free words of length less than n Introduction In general the problem of evaluating the number un of words of length n in the language U consisting of words on some nite alphabet A with no factors in a certain set F is not easy If F is nite it amounts to counting words in the rational language U and there are well known techniques for this e g computing the generating function But if for instance F is the set of images of a pattern p by non erasing morphisms it is not even known how to decide whether U is nite or in nite i e whether p is unavoidable or avoidable on A However Brandenburg proved that un grows exponentially for the pattern p on a ternary alphabet A then U is the set of square free ternary words and Goralcik and Vanicek proved the same for any avoidable binary pattern p on a binary alphabet A also proved by Brandenburg for p Here we shall study the growth of un in the case where F is the set F f xuxux j x A and u A g with A fa bg the elements of U are then called overlap free binary words After recalling what is known about un end of this section we describe Section a bijection between U and a rational language From the automaton recognizing this language we construct Section explicit recurrence relations veri ed by un showing that un can be computed in logarithmic time In Section we study the consequences of these relations on the asymptotic behaviour of un In particular we prove that although un is bounded from below and from above by polynomial quantities it is not itself equivalent to a polynomial Overlap Free Binary Words We consider words on the alphabet A fa bg i e elements of the monoid A The letter denotes the empty word We shall use the notation a b and b a A word w contains an overlap if some factor v appears at two overlapping positions It is equivalent to say that w contains a factor of the form xuxux with x A and u A A word w is called overlap free if it contains no overlap In the language of patterns an overlap free word is a word which simultaneously avoids the patterns and Previous Results In Axel Thue proved that there are in nitely many overlap free binary words or equivalently there are arbitrarily long overlap free binary words To do this he constructed an in nite overlap free word a where is the morphism