Journal Article10.1017/S0963548302005187
Pattern Avoidance and Overlap in Strings
TL;DR: If cor (w) > cor(w′) then aw (n)−aw′(n) > (|Ω|−1)(aw(n−1)−AW′( n−1)) for n [ges ] N, and the value of N is given, confirming a conjecture by Eriksson.
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Abstract: Consider a finite alphabet Ω and patterns which consist of characters from Ω. For a given pattern w, let cor(w) denote its autocorrelation, which can be seen as a measure of the amount of overlap in w. Letting aw(n) denote the number of strings over Ω of length n which do not contain w as a substring, the main result of this paper is: If cor(w) > cor(w′) then aw(n)−aw′(n) > (|Ω|−1)(aw(n−1)−aw′(n−1)) for n ≥ N, and the value of N is given. This result confirms a conjecture by Eriksson [2], which was previously proved to be true by Cakir, Chryssaphinou and Mansson [1] when |Ω| ≥ 3.
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References
On a Conjecture by Eriksson Concerning Overlap in Strings
TL;DR: It is proved that ∣Ω∣≥3, by giving a lower bound for aw(n)−aw′(n), which is the number of strings of length n that do not contain w as a substring.
On Pattern Frequency Occurrences in a Markovian Sequence
TL;DR: exact and asymptotic formulae for moments, probability of r pattern occurrences for three different regions of r, namely: (i) r=O(1) , (ii) central limit regime, and (iii) large deviations regime are presented.
On pattern frequency occurrences in a Markovian sequence
Mireille Régnier,Wojciech Szpankowski +1 more
- 29 Jun 1997
TL;DR: In this article, the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately was studied, and the results for the Markovian model were summarized in two theorems.
Autocorrelation and the Enumeration of Strings Avoiding a Fixed String
TL;DR: A simple criterion on the autocorrelations of w and w′ is given for determining whether aw(n) > aw′(n), which is likely to be large enough for all large enough n.
String overlaps, pattern matching, and nontransitive games
TL;DR: The key notion of the correlation of two strings is introduced, which is a representation of how the second string can overlap into the first, and this notion is used to state and prove a formula for the generating function that enumerates the q -ary strings of length n which contain none of a given finite set of patterns.