Generalized reflection complexity

@misc{blazek2026generalized,
  title = {{Generalized reflection complexity}},
  author = {Adam Blažek and Lubomíra Dvořáková and Edita Pelantová},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14637},
  eprint = {2610.14637},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14637},
}

TY  - UNPB
TI  - Generalized reflection complexity
AU  - Adam Blažek
AU  - Lubomíra Dvořáková
AU  - Edita Pelantová
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14637
UR  - https://arxiv.org/abs/2610.14637
AB  - 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 𝒜. The G-complexity of a sequence over 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.
ER  -