Effective computations of abelian complexities

In combinatorics on words, a conjecture due to Parreau et al. in 2015 claims that, given an abstract numeration system S, the k-abelian complexity (a generalization of the abelian complexity due to Karhumäki et al. in 2013) of an S-automatic sequence is itself S-regular. Some isolated instances for this conjecture have been identified, but no large family of sequences. In this talk, such a family of sequences is exhibited, which are fixed points of Pisot-type substitutions and for which the abstract numeration system is the classical Dumont-Thomas numeration system associated with the substitution. This is a joint work with J.-M. Couvreur (Orléans, France), M. Delacourt (Orléans, France), N. Ollinger (Orléans, France), P. Popoli (Liège, Belgium), and J. Shallit (Waterloo, Canada).