Extensions of rich words

In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The unified study of words which achieve this limit was initiated in [A. Glen et al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these words rich (in palindromes). This article contains several results about rich words and especially extending them. We say that a rich word w can be extended richly with a word u if wu is rich. Some notions are also made about the infinite defect of a word, the number of rich words of length n and two-dimensional rich words.Comment: 19 pages, 3 figure

The most unbalanced words 0q−p1p and majorization

A finite word w &isin; {0, 1}&lowast; is balanced if for every equal-length factors u and v of every cyclic shift of w we have ||u|1 &minus; |v|1| &le; 1. This new class of finite words was defined in [O. Jenkinson and L. Q. Zamboni, Characterisations of balanced words via orderings, Theoret. Comput. Sci. 310(1&ndash;3) (2004) 247&ndash;271]. In [O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl. 1(4) (2009) 463&ndash;484], there was proved several results considering finite balanced words and majorization. One of the main results was that the base-2 orbit of the balanced word is the least element in the set of orbits with respect to partial sum. It was also proved that the product of the elements in the base-2 orbit of a word is maximized precisely when the word is balanced. It turns out that the words 0q&minus;p1p have similar extremal properties, opposite to the balanced words, which makes it meaningful to call these words the most unbalanced words. This paper contains the counterparts of the results mentioned above. We will prove that the orbit of the word u = 0q&minus;p1p is the greatest element in the set of orbits with respect to partial sum and that it has the smallest product. We will also prove that u is the greatest element in the set of orbits with respect to partial product.</p

On Kaluza’s sign criterion for reciprocal power series

T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzyż is applied

On Kaluza's sign criterion for reciprocal power series

T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzy\.z is applied.Comment: 13 page

On a question of Hof, Knill and Simon on palindromic substitutive systems

International audienceIn a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the subshift Ω of the potential of a discrete Schrödinger operator which guarantees purely singular continuous spectrum on a generic subset of Ω. In part, this condition requires that the subshift Ω be palindromic, i.e., contains an infinite number of distinct palindromic factors. In the same paper, they introduce the class P of morphisms f : A * → B * of the form a → pq a with p and q a palindromes, and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form a → q a p. In this paper we give a partial affirmative answer to the question of Hof, Knill and Simon: we show that every rich primitive substitutive subshift is generated by at most two morphisms each of which is conjugate to a morphism of class P. More precisely, we show that every rich (or almost rich in the sense of finite defect) primitive morphic word y ∈ B ω is of the form y = f (x) where f : A * → B * is conjugate to a morphism of class P, and where x is a rich word fixed by a primitive substitution g : A * → A * conjugate to one in class P