Enumerating Abelian Returns to Prefixes of Sturmian Words

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 ${\mathcal{A}\mathcal{P}\mathcal{R}}_u$ 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 ${\mathcal{A}\mathcal{P}\mathcal{R}}_u$ and we determine it for the characteristic Sturmian words.