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arXiv · 2405.18032

Automatic Abelian Complexities of Parikh-Collinear Fixed Points

Michel Rigo·Manon Stipulanti·Markus A. Whiteland
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Abstract

Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT-WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set.

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Michel Rigo, Manon Stipulanti, Markus A. Whiteland. 2024-05-28. Automatic Abelian Complexities of Parikh-Collinear Fixed Points. https://arxiv.org/abs/2405.18032

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