Abstract numeration systems on bounded languages and multiplication by a constant

Emilie Charlier (1) , Michel Rigo (1) , Wolfgang Steiner (2)

1 Département de Mathématiques [Liège]
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Emilie Charlier, Michel Rigo, Wolfgang Steiner. Abstract numeration systems on bounded languages and multiplication by a constant. Integers : Electronic Journal of Combinatorial Number Theory, 2008, 8 (1), pp.#35. ⟨hal-00151512v2⟩