LR property of non-well-formed scales

Abstract

This article studies generated scales having exactly three different step sizes within the language of algebraic combinatorics on words. These scales and their corresponding step-patterns are called non well formed. We prove that they can be naturally inserted in the Christoffel tree of well-formed words. Our primary focus in this study is on the left- and right-Lyndon factorization of these words. We will characterize the non-well-formed words for which both factorizations coincide. We say that these words satisfy the LR property and show that the LR property is satisfied exactly for half of the non-well-formed words. These are symmetrically distributed in the extended Christoffel tree. Moreover, we find a surprising connection between the LR property and the Christoffel duality. Finally, we prove that there are infinitely many Christoffel–Lyndon words among the set of non-well-formed words and thus there are infinitely many generated scales having as step-pattern a Christoffel–Lyndon word.

Keywords:

• well-formed scales
• non-well-formed words
• Three Distance Theorem
• Lyndon words
• Christoffel words
• generated scales
• standard factorization

2010 Mathematics Subject Classification:

• 00A65
• 68R15

2012 Computing Classification Scheme:

• Combinatorics on words

Acknowledgements

The authors thank the Editors-in-Chief, Clifton Callender and Thomas Fiore, as well as the anonymous referees, for their suggestions and remarks, which have improved the final presentation of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.