Generalized reflection complexity

The reflection complexity of a sequence, introduced recently by Allouche, Campbell, Li, Shallit, and Stipulanti, counts the factors of a given length up to reversal. We generalize this notion considering a group $G$ generated by antimorphisms over a given alphabet $\mathcal{A}$. The $G$-complexity of a sequence over $\mathcal{A}$ counts the factors of a given length up to the action of $G$. For $G$ generated by the reflection alone, the $G$-complexity coincides with the reflection complexity. We characterize eventual periodicity in terms of the $G$-complexity. Then we focus on the binary alphabet and the group $H$ generated by reflection and the antimorphism exchanging letters. We provide a simple formula for the $H$-complexity in the case of $H$-rich sequences and determine the precise values for complementary symmetric Rote sequences and generalized binary Thue-Morse sequences. Finally, we describe the aperiodic recurrent sequences having the minimum $H$-complexity – they are exactly the complementary symmetric Rote sequences, while the binary sequences with the minimum reflection complexity correspond to the Sturmian sequences.