Geodesic
A new approach to the 2-regularity of the \(\ell\)-abelian complexity of 2-automatic sequences
Aline Parreau1 ; Michel Rigo2 ; Eric Rowland2 ; Élise Vandomme3
1Lyon University - CNRS
2Liege University
3Grenoble University Liege University
The electronic journal of combinatorics, Tome 22 (2015) no. 1
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We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-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.
DOI : 10.37236/4478
Classification : 68R15, 05A05, 11B85
Mots-clés : automatic sequences, abelian complexity, regular sequences, Thue-Morse word, period-doubling word

Aline Parreau  1   ; Michel Rigo  2   ; Eric Rowland  2   ; Élise Vandomme  3

1 Lyon University - CNRS
2 Liege University
3 Grenoble University Liege University 
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     title = {A new approach to the 2-regularity of the \(\ell\)-abelian complexity of 2-automatic sequences},
     journal = {The electronic journal of combinatorics},
     year = {2015},
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Aline Parreau; Michel Rigo; Eric Rowland; Élise Vandomme. A new approach to the 2-regularity of the \(\ell\)-abelian complexity of 2-automatic sequences. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4478

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