Sturmian words and the Stern sequence

TL;DR

This paper explores the deep connections between Sturmian words, Christoffel and central words, and the Stern sequence, revealing new combinatorial insights and a non-commutative version of a known theorem, with implications for word length distributions.

Contribution

It introduces a novel interpretation of Stern's sequence in terms of Christoffel and central words, and presents a non-commutative version of the alternating bit sets theorem.

Findings

Interpretation of Stern's sequence as lengths and periods of specific words

New bounds and inequalities for lengths of Christoffel words

A non-commutative generalization of the alternating bit sets theorem

Abstract

Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart of characteristic Sturmian words. A natural arithmetization of the theory is obtained by representing central and Christoffel words by irreducible fractions labeling respectively two binary trees, the Raney (or Calkin-Wilf) tree and the Stern-Brocot tree. The sequence of denominators of the fractions in Raney's tree is the famous Stern diatomic numerical sequence. An interpretation of the terms $s(n)$ of Stern's sequence as lengths of Christoffel words when $n$ is odd, and as minimal periods of central words when $n$ is even, allows one to interpret several results on Christoffel and central words in terms of Stern's sequence and, conversely, to obtain a new insight in the combinatorics of Christoffel and central words by using properties of Stern's sequence. One of our main results is a non-commutative version of the "alternating bit sets theorem" by Calkin and Wilf. We also study the length distribution of Christoffel words corresponding to nodes of equal height in the tree, obtaining some interesting bounds and inequalities.