Complexity function: Difference between revisions

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A set ''S'' of finite binary words is ''balanced'' if for each ''n'' the subset ''S''_''n'' of words of length ''n'' has the property that the [[Hamming weight]] of the words in ''S''_''n'' takes at most two distinct values. A '''balanced sequence''' is one for which the set of factors is balanced.<ref name=AS313>Allouche & Shallit (2003) p.313</ref> A balanced sequence has complexity function at most ''n''+1.<ref name=L48>Lothaire (2011) p.48</ref>

A [[Sturmian word]] over a binary alphabet is one with complexity function ''n''&nbsp;+&nbsp;1.<ref name=PF6>Pytheas Fogg (2002) p.6</ref> A sequence is Sturmian if and only if it is balanced and aperiodic.<ref name=L46>Lothaire (2011) p.46</ref><ref name=AS318>Allouche & Shallit (2003) p.318</ref> An example is the [[Fibonacci word]].<ref name=PF6/><ref name=deluca1995>{{cite journal | journal=Information Processing Letters | year=1995 | pages=307–312 | doi=10.1016/0020-0190(95)00067-M | title=A division property of the Fibonacci word | first=Aldo | last=de Luca | volume=54 | issue=6 }}</ref> More generally, a Sturmian word over an alphaberalphabet of size ''k'' is one with complexity ''n''+''k''−1. An Arnoux-Rauzy word over a ternary alphabet has complexity 2''n''&nbsp;+&nbsp;1:<ref name=PF6/> an example is the [[Tribonacci word]].<ref name=PF368>Pytheas Fogg (2002) p.368</ref>

For [[recurrent word]]s, those in which each factor appears infinitely often, the complexity function almost characterises the set of factors: if ''s'' is a recurrent word with the same complexity function as ''t'' are then ''s'' has the same set of factors as ''t'' or δ''t'' where δ denotes the letter doubling morphism ''a'' → ''aa''.<ref name=BLRS84>Berstel et al (2009) p.84</ref>