Résumé :

A numeration system is a way to represent numbers by words that is, sequence of digits. For instance, the number of letters in this talk's title is represented by the word 29 in the usual base-10 numeration system, by the string 11101 in binary (29=16+8+4+1), by 1010000 in the Fibonacci numeration system (29=21+8), and by 21011021 in the base-3/2 numeration system. Once numbers are represented by words, numeration systems may be studied from the perspective of formal languages.
This talk is about the notion of breadth-first signature. It is a way to serialize ordered infinite tree of finite degree into an infinite word of integers. It can be extended to prefix-closed languages and numeration systems. We will see that properties of the signature provide insights about the numeration systems. For instance, the signature of the abstract numeration systems (Lecomte Rigo) form a special subclass of substitutive words. Similarly, signatures that are ultimately periodic are linked to integer base numeration systems or rational base numeration systems (Akiyama, Frougny and Sakarovitch).