Enumerating Abelian Returns to Prefixes of Sturmian Words

TL;DR

This paper investigates the set of abelian returns to prefixes of Sturmian words, providing formulas, algorithms, and characterizations based on continued fraction coefficients and intercepts.

Contribution

It introduces a method to determine the cardinality and structure of abelian returns for Sturmian words, extending previous work with new algorithms and explicit characterizations.

Findings

Cardinality of abelian returns expressed via continued fraction coefficients

Algorithm for computing the set of abelian returns

Explicit description for characteristic Sturmian words

Abstract

We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set $\mathcal{APR}_u$ of abelian returns of all prefixes of a Sturmian word $u$ in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set $\mathcal{APR}_u$ and we determine it for the characteristic Sturmian words.