Shuffle of Words and Araucaria Trees

Issue title: SPECIAL ISSUE MCU2004

Article type: Research Article

Authors: Schott, René | Spehner, Jean-Claude

Affiliations: LORIA and IECN, Université Henri Poincaré, 54506 Vandoeuvre-lès-Nancy, France. E-mail: schott@loria.fr | Laboratoire MIA, Equipe MAGE, FST, Université de Haute Alsace, 68093, Mulhouse, France. E-mail: JC.Spehner@uha.fr

Abstract: The shuffle of k words u_1,…,u_k is the set of words obtained by interleaving the letters of these words such that the order of appearance of all letters of each word is respected. The study of the shuffle product of words leads to the construction of an automaton whose structure is deeply connected to a family of trees which we call araucarias. We prove many structural properties of this family of trees and give some combinatorial results. We introduce a family of remarkable symmetrical polynomials which play a crucial role in the computation of the size of the araucarias. We prove that the minimal partial automaton which recognizes the shuffle of a finite number of special words contains an araucaria for each integer k>0.

Keywords: Automaton, shuffle of words, remarkable polynomials, trees

Journal: Fundamenta Informaticae, vol. 74, no. 4, pp. 579-601, 2006