Second order balance property on Christoffel words

Lama Tarsissi (1) , Laurent Vuillon (1)

1 LAMA - Laboratoire de Mathématiques

In this paper we study the balance matrix that gives the order of balance of any binary word. In addition, we define for Christoffel words a new matrix called second order balance matrix. This matrix gives more information on the balance property of a word that codes the number of occurrences of the letter 1 in successive blocks of the same length for the studied Christoffel word. By taking the maximum of the Second order balance matrix we define the second order of balance and we are able to order the Christoffel words according to these values. Our construction uses extensively the continued fraction associated with the slope of each Christoffel word, and we prove a recursive formula based on fine properties of the Stern-Brocot tree to construct second order matrices. Finally, we show that an infinite path on the Stern-Brocot tree, which minimizes the second order of balance is given by a path associated with the Fibonacci word.

Mots clés: Balance property; Second order balance property; Christoffel words; SternBrocot tree; Continued fractions

Domaines: Systèmes dynamiques [math.DS]; Mathématique discrète [cs.DM]

Lama Tarsissi, Laurent Vuillon. Second order balance property on Christoffel words. Daniel Slamanig; Elias Tsigaridas; Zafeirakis Zafeirakopoulos. Mathematical Aspects of Computer and Information Sciences. 8th International Conference, MACIS 2019, Gebze, Turkey, November 13–15, 2019, Revised Selected Papers, 11989, Springer, pp.295-312, 2020, Lecture Notes in Computer Science, 978-3-030-43119-8. ⟨10.1007/978-3-030-43120-4_23⟩. ⟨hal-02433984v2⟩