Article (Scientific journals)
Automatic sequences based on Parry or Bertrand numeration systems
Massuir, Adeline; Peltomäki, Jarkko; Rigo, Michel
2019In Advances in Applied Mathematics, 108, p. 11-30
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Keywords :
Numeration system; Parry number; Bertrand theorem; Combinatorics on words; Factor complexity; automata
Abstract :
[en] We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical k-automatic sequences and Pisot-automatic sequences. We show that, like k-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for k-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is U -automatic with respect to a positional numeration system U with regular language of numeration if and only if its U -kernel is finite.
Disciplines :
Mathematics
Computer science
Author, co-author :
Massuir, Adeline ;  Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Peltomäki, Jarkko
Rigo, Michel  ;  Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Language :
English
Title :
Automatic sequences based on Parry or Bertrand numeration systems
Publication date :
2019
Journal title :
Advances in Applied Mathematics
ISSN :
0196-8858
eISSN :
1090-2074
Publisher :
Elsevier, Atlanta, Georgia
Volume :
108
Pages :
11-30
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBi :
since 29 March 2019

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