A characterization of the words occurring as factors in a minimum number of words

AbstractWe show by an injective proof that a word w of length k⩾2 occurs as a factor in a minimum number of words of length n(n>k) if and only if all letters of w are equal

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

AbstractIn 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 (a≠b) is deduced. Let kn 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 limsupn→∞k(n)/lnn<1/(3lnm) then almost all words of length n over A contain the factor (aba)kn as n→∞. Using the properties of the roots of the recurrence satisfied by L(n) it is shown that if limsupn→∞k(n)/lnn>1/(3lnm) then this property is false. Moreover, if limn→∞(lnn-3klnm)=η∈R then limn→∞|W(n,(aba)kn,A)|/mn=1-exp(-(1-1/m3)exp(η)), where W(n,(aba)kn,A) denotes the set of words of length n over A containing the factor (aba)kn

On words containing all short subwords

AbstractLutz Priese raised the following conjecture: Almost all words of length n over a finite alphabet A with m letters contain as subwords all words of length ⌊log log n⌋ over A as n → ∞. In this note we prove that this property holds for subwords of length k(n) over A provided limn → ∞ k(n)log n = 0