Note: On the number of words containing the factor (aba)k

  • Authors:

  • Ioan Tomescu

  • Affiliations:

  • Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei, 14, 010014 Bucharest, Romania

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

In this paper a recurrence relation satisfied by the number L(n) of words of length n over an alphabet A of cardinality m(m=2) not containing the factor (aba)^k(ab) is deduced. Let k"n be a sequence of positive integers. From [I. Tomescu, A threshold property concerning words containing all short factors, Bull. EATCS 64 (1998) 166-170] it follows that if limsup"n"-"~k(n)/lnn~. Using the properties of the roots of the recurrence satisfied by L(n) it is shown that if limsup"n"-"~k(n)/lnn1/(3lnm) then this property is false. Moreover, if lim"n"-"~(lnn-3klnm)=@h@?R then lim"n"-"~|W(n,(aba)^k^"^n,A)|/m^n=1-exp(-(1-1/m^3)exp(@h)), where W(n,(aba)^k^"^n,A) denotes the set of words of length n over A containing the factor (aba)^k^"^n.