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In [[mathematics]], a '''locally catenative sequence''' is a sequence of [[word (mathematics)|words]] in which each word can be constructed as the concatenation of previous words in the sequence.<ref>{{cite book
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  | last = Rozenberg
  | first = Grzegorz
  | coauthors = Salomaa, Arto
  | title = Handbook of Formal Languages
  | publisher = Springer
  | date = 1997
  | pages = 262
  | isbn = 3-540-60420-0}}</ref>


Formally, an infinite sequence of words ''w''(''n'') is locally catenative if, for some positive integers ''k'' and ''i''<sub>1</sub>,...''i''<sub>''k''</sub>:
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:<math>w(n)=w(n-i_1)w(n-i_2)...w(n-i_k) \text{ for } n \ge \max(i_1, ... i_k) \, .</math>
 
Some authors use a slightly different definition in which encodings of previous words are allowed in the concatenation.<ref>{{cite book
  | last = Allouche
  | first = Jean-Paul
  | coauthors = Shallit, Jeffrey
  | title = Automatic Sequences
  | publisher = Cambridge
  | date = 2003
  | pages = 237
  | isbn = 0-521-82332-3}}</ref>
 
==Examples==
The sequence of [[Fibonacci word]]s ''S''(''n'') is locally catenative because
:<math>S(n)=S(n-1)S(n-2) \text{ for } n \ge 2 \, .</math>
 
The sequence of [[Thue-Morse sequence|Thue-Morse words]] ''T''(''n'') is not locally catenative by the first definition. However, it is locally catenative by the second definition because
:<math>T(n)=T(n-1)\mu(T(n-1)) \text{ for } n \ge 1 \, ,</math>
where the encoding &mu; replaces 0 with 1 and 1 with 0.
 
==References==
<references/>
 
[[Category:Formal languages]]
[[Category:Combinatorics on words]]

Latest revision as of 01:52, 12 January 2015

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