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Discrete Mathematics & Theoretical Computer Science |
Frédérique Bassino ; Julien Clément ; Julien Fayolle ; Pierre Nicodème - Constructions for Clumps Statistics.
dmtcs:3563 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, DMTCS Proceedings vol. AI, Fifth Colloquium on Mathematics and Computer Science - https://doi.org/10.46298/dmtcs.3563
1,2; Julien Clément 3; Julien Fayolle 4; Pierre Nicodème 5
- 1 Laboratoire d'Informatique de Paris-Nord
- 2 Laboratoire d'Informatique Gaspard-Monge
- 3 Equipe AMACC - Laboratoire GREYC - UMR6072
- 4 Laboratoire d'Informatique de Paris-Nord
- 5 Laboratoire d'informatique de l'École polytechnique [Palaiseau] [LIX]
We consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences.