Séries rationnelles et distributions de longueurs

Abstract : This work concerns rational series having nonnegative integral coefficients and is centered on two kinds of questions : the star-height problem and the study of the properties of length distributions of codes. We study the star-height problem in the case of rational series of a particular kind : N-rational series in one variable. We characterize in various ways N-rational series having star-height 1. We also give a criterion for deciding the star-height of an important class of N-rational series in one variable. Basically, the study of the star-height of the N-rational series in one variable makes use of the properties of their representations by matrices. We establish in particular, from a result of Handelman, a characterisation of the spectral radius of an irreducible companion matrix having nonnegative integral entries. Next, we study length distributions of circular codes and of prefix codes. We prove three new results about circular codes. We generalize in several directions the characterization of length distributions of circular codes established, by Schützenberger, in the case of a finite alphabet. On the one hand, we replace the finite alphabet by an arbitrary alphabet whose elements are weighted ; this allows us to extend the result to two length distributions. On the other hand, we restrict the conditions which allows us to establish the decidability in the case of a finite sequence. We give a new formulation of this characterization. This result established by combinatorial methods underscores the decidability in the case of a finite distribution. We establish a necessary and sufficient condition for a sequence of nonnegative integers to be the length distribution of a maximal circular code over a finite alphabet. Finally, we emphasize the links between the generating series of rational prefix codes and a class of N-rational series : the DOL-series. We give a sufficient condition for an N-rational sequence to be the length distribution of a maximal rational prefix code over a k-letter alphabet.
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Frédérique Bassino. Séries rationnelles et distributions de longueurs. Combinatoire [math.CO]. Université de Marne la Vallée, 1996. Français. ⟨tel-00719366⟩

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