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A new approach to the 2-regularity of the -abelian complexity of 2-automatic sequences

Aline Parreau 1, 2 Michel Rigo 1 Eric Rowland 1 Elise Vandomme 1
2 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem to develop a general approach for studying the-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue–Morse word have 2-abelian complexity sequences that are 2-regular. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular.
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Contributor : Aline Parreau <>
Submitted on : Wednesday, April 22, 2015 - 3:00:30 PM
Last modification on : Tuesday, June 1, 2021 - 2:08:09 PM
Long-term archiving on: : Wednesday, April 19, 2017 - 3:08:00 AM


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  • HAL Id : hal-01144698, version 1
  • ARXIV : 1405.3532


Aline Parreau, Michel Rigo, Eric Rowland, Elise Vandomme. A new approach to the 2-regularity of the -abelian complexity of 2-automatic sequences. The Electronic Journal of Combinatorics, Open Journal Systems, 2015, 22 (1), pp.#P1.27. ⟨hal-01144698⟩



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