Constructions for Clumps Statistics

Abstract: 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 parameter 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 emphasize the fact that, in contrast with the probabilistic approach which only provides asymptotic results, the combinatorial approach provides exact results that are useful when considering short sequences.