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Shuffle of Words and Araucaria Trees

100%

Schott R. , Spehner J.-C.

2006

tom Vol. 74, nr 4

579-601

The shuffle of k words u1,..., uk 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.

Araucaria Trees: Construction and Grafting Theorems

88%

Schmitt D. , Spehner J.-C.

tom Vol. 123, nr 4

417--445

Araucarias have been introduced by Schott and Spehner as trees which appear in the minimal automaton of the shuffle of words. We give here a new definition of araucarias which is more constructive and we prove that our definition of araucarias is equivalent to the original one. From the new definition we derive an optimal algorithm for the construction of araucarias and a new method for calculating their size. Moreover we characterize araucarias by properties of their maximal paths, by associating a capacity to every edge. We then show that every araucaria can be obtained by grafting and merging smaller araucarias. We prove also that every directed tree can be embedded in an araucaria. Moreover we define a capacity for every vertex of an araucaria, which leads to different new enumeration formulas for araucarias.